While both his parents have British accents, Daniel has an American accent because he was raised in America (possibly Massachusetts); he only studied in England. After being booted out of Oxford, he returned to his home, where he began living with a caretaker. Also, only one parent raised him (Eloise) and since she didn't seem to be around much, the influence of her accent on Daniel would be weakened even more.
Being raised in the U.S. would have influenced Daniel's accent more than that of his mother. The seminal work on this phenomenon (William Labov, Sociolinguistic Patterns, Univ. of Pennsylvania Press, 1973) details research that demonstrates that as children leave home for school, they are more influenced by the linguistic patterns of their peers than of their parents.
The reason Eloise would have raised Daniel in America and not England would have been her estrangement from Charles Widmore; since he and Penny were in England, Eloise would have wanted to be as far away from them as possible.
Widmore was forced off the island at some point after the Purge, when he was much older. So he was not living permanently in England until Daniel was in his late teens or early 20s.
Faraday was raised in America, but Eloise wasn't the one who raised him. As he was raised in America, possibly in the Northeastern United States (Massachusetts, New York, Maine, Vermont, etc) he picked up an American accent. Given that Faraday could shift through time, he at some unkown point, shifted to an earlier point in time, and learnt to play Fantaisie-Impromptu in C Sharp and use it as some sort of a memory aid.
Eloise could have raised him. Children born of immigrant parents with accents do not develop accents themselves (although they can usually do excellent impressions of their parents' accents). Faraday has no sign of an accent from the Northeast. Jeremy Davies was born in Michigan and grew up in Kansas and California. It is possible to detect the origin of some of the actors if you are skilled in linguistics. Terry O'Quinn (Locke) is clearly from the Midwest even though Locke grew up in California, Evangeline Lilly's Canadian accent faintly comes through at certain points, and Ken Leung screws up his line in "Some Like It Hoth" when he says he was waiting "on line" at the cafeteria instead of "in line," something nobody his age from Encino would ever say.
Where the name Faraday came from
The name "Faraday" did not come from his mother, Ms. Hawking, or his father, Mr. Widmore. However, Ms. Hawking named her son after the person she shot in the jungle; she would have presumably checked his wallet and ID after he died. Thus, Daniel Faraday was the first person to be named after himself.
Eloise gave Daniel the Faraday surname, because she heard Sayid calling him that in 1977. It was crucial for Dan's destiny.
Eloise Hawking married a man named Faraday after leaving the Island and both mother and son took his name. After splitting with him, Eloise reverted back to Hawking whilst Daniel kept Faraday.
Eloise didn't want Daniel to know his father was Widmore, due to his infidelity when they were together and that he is a powerful and dangerous man. She likely made up a story and a last name to keep Daniel from knowing the true identity of his father. We won't find out why his last name is Faraday though, as it isn't really important to the story.
Eloise's full name as of the 1977 events in "Follow the Leader" is "Eloise Faraday." Sometime after "Follow the Leader", Eloise leaves the Island and gives birth to Daniel, giving him the name Daniel Faraday. Some time later, she marries a man with the last name "Hawking," making her "Eloise Hawking."
Eloise changed his name and told him that it was his father's name who had passed away.
After Eloise killed Daniel, she read through his journal and saw that his notes on time travel, the Jughead and whatever else he had in there and realized that his breakthroughs in science would be vital to the events taking place on the island. She pushed Daniel toward 100% devotion to science his entire life (no time for music or relationships), so when he was born, she named him Faraday (after the famous Physicist Michael Faraday) to further push him toward a life of science.
We know that much of the information in his journal is unclear to us how it got there. We also know that Daniel tested his invention on himself and that experiment created his memory loss. Now that Daniel is dead, this is the only event we know about that could have caused him to gather information for his journal that he hadn't experienced in the present. This testing caused him to travel in time (either physically, or just his mind) and traveling is when he gathered all the information we see in his journal. We know that the test on Theresa caused her to go into a coma, leading us to believe that only his mind was traveling in time. His mind would have had to travel forward in time (unlike Desmond), and after each flash, he wrote the experience in the journal that his mother gave him after he graduated from Oxford.
The journal is new when it's given to Daniel as a graduation gift. After receiving the journal, Daniel begins his research with Eloise the rat. A good scientist writes down all scientific observations in a notebook or journal. He sends Eloise the rat's mind to the future briefly, and she knows how to run the maze. Then he starts experimenting on himself. He sends his mind to the future on the Island (DHARMA time) and writes his observations in his journal. He does this many times, and has a journal full of bits and pieces from the future on the island. He experiments on himself too many times and fries his memory.
Daniel says his journal contains "everything he's learned about the DHARMA Initiative." He had this information before he came to the Island on the freighter. (See his notations about The Orchid in Season 4 as an example.) That means he must have traveled there previously, sending himself back in time during his experiments. How else would he know that the Incident was about to occur that afternoon? It doesn't appear that Eloise gives Dan the journal with the notes. She gives him a blank journal that he proceeds to fill with notes about the DHARMA Initiative as he conducts his experiments, but he forgets what he has learned and must continually refer to the journal for guidance.
Eloise takes the journal from Daniel after she shoots him. She uses the journal to know what is going to happen from the day Daniel graduates to the day she sends him back to the Island to die.
She does take his journal after his death, but does not give him it back as his graduation present. She buys him a new book that would become this journal as he does his experiments. It's like a self-fulfilling prophecy; Eloise gave him the journal, and the information he recorded in it led him to the Island and his death. His death led Eloise to take the journal and therefore knew to buy him a journal in the first place. It's one big loop.
You know, I always just figured he googled "DHARMA Initiative" and copied that into his notebook. I figured he did research on the DHARMA Initiative while performing his experiments. They both studied time and I feel Daniel was smart enough to be able to get his hands on DHARMA information. I don't think Daniel went into the future, I believe when he tested himself he just created memory loss instead of the desired effect of experiencing the future.
Contains detail of multiple time lines, and will be able to differentiate certain events to come in that certain time line.
Faraday's time loop is like Locke's
Locke is seen as special by the others in large part because he told Richard and Widmore in 1954 that he would be their leader. Locke has backed it up by showing that he is different, but he may have never been brought to the Island at all if not for Locke himself speaking to the others in 1954. In that same vein, Faraday is in a loop because his mother raised him to be a scientist and will send him off to die. However, this is only due to the fact that an adult Faraday dies at her feet and leaves a journal predicting this path.
When he was born
He was born after Eloise shot him. In Follow The Leader, while Jack is asking Richard who Widmore is, there is a scene of Widmore and Eloise talking and he is touching her stomach, indicating his concern for their unborn child: Faraday.
Eloise was pregnant in 1977. But as a result of a time travel incident we haven't yet seen, she will give birth to Daniel years earlier.
There is nothing to suggest that Eloise time travelled and gave birth to Daniel years earlier. However the fact that he has a birth certificate indicates that she somehow left the island between the timing of the Incident and his birth in the US.
When Richard & Eloise accompany Jack & Sayid to remove the core from the bomb, Richard comments that she is pregnant.
Daniel could not have been born in 1977, because that would mean he was only 19 when he was a professor at Oxford in 1996 - unlikely, even considering his advanced intellect.
It's very possible and actually happens. There are child progidies who earn PhDs in their teens. Google "youngest college graduates" - Sho Yano got 1500/1600 on his SAT at 9 years old, earned a PhD at 18 and then an MD at 21. The youngest professor in MIT history was 20. Daniel states that he was the "youngest doctor to ever graduate from Oxford" (which means that, if the LOST writers did their research, he was younger than Ruth Lawrence when she graduated, who holds the record at Oxford for earning a Master's at 15 and PhD at 17). He could have been 16 or 17 when he graduated and been an assistant professor at 19. If so, they would have been eager to have a genius like him stay there to do his research, when he would have been sought by every university in the world. So while this would have been very impressive, it's not in the realm of unbelievable.
He knew he was going to get shot/die
Daniel's mother kept his "full" diary, the one she took after killing him. That's how she was able to know the future. It is therefore possible that Daniel found this "full" diary at some point in the future and read it. He then deduced that he will die on the day of his last entry, or maybe because it is written there, perhaps by his mother.
In "The Variable", notice very closely his behavior before being shot... he counts 1, 2, then pauses as if he is waiting for the shot. It happens very quickly. Also, he told Kate and Jack earlier that he was a beginner with guns, so there is no way he was going to shoot Richard. Also, after being shot, he was shocked only because it was his own mother that shot him, not because he was shot.
Daniel did not know when he was going to die as he did not diarise his own death due to his... death. That note was never left in his diary as evidenced by his shock when he discovers his own mother sent him back to the island knowing he was going to die.
He had realized that he could die once he considered "the variables" that meant any of them could die, but that doesn't mean he was sure he would. If he had known he was going to die but not how, then his shock would have been that his mother was the one who killed him. He would have replied with something to that effect, e.g. "It was you!" rather than "You knew all along!"
Discovering Desmond is Special
Faraday, while having problems with his memory, told Desmond to meet him in Oxford and tell him the settings for his machine, because he figures that this is what has always happened. However, when he gets his memory back he remembers that this isn't the case. Desmond has somehow managed to cause a change in the past.
He tells Desmond to find Eloise, but why? She already knew he was on the island. In actuality, this was Faraday's way of alerting Hawking to Desmond's importance. Daniel asked for her specifically because there's some time voodoo going on. Since Desmond is able to communicate with Faraday, he must be able to get around the restrictions of time travel.
Daniel is the 'clever man' who created the Pendulum
Daniel went to Ann Arbor on the sub Galagainstead of staying on the island in 1974. I theorized somehow he found out about the Lamp Post and then went on to California to the church. (Although, actually going to the lamp post is not even necessary. He could have solved the problem and given it to his mother somehow to have it installed later.) It has been said that the lamp post was built by 1970 for directing the sub by beacon. If the beacon was no longer functioning some other method would be needed to re-locate the island. The pendulum needed to be in place for the future when the beacon was not transmitting. Faraday knew this and that is why he went to solve this problem.
The formulas in the lamp post look like the writing in his journal. 'Finding where the island is not' is a very typical genius thought that Faraday would have.
Note that Eloise had his journal since 1977 and certainly guided Daniel's destiny based on this. At some point (after the purge - and I'm leaning towards the 1992 date) she took over the lamp post from Darhma. When she says 'a very clever man' I think she says it as with a bit of pride knowing it was Daniel.
Someone at the DHARMA Initiative created the Lamp Post prior to 1974, in order to find the island in the first place. The pendulum is what always found the island, so as long as the pendulum worked, they could find it again. The equations there looked like Daniel's equations because they were both the same type of equations, involving extremely complicated theories of quantum physics.