Since Then: Carole didn't know about Aaron for the first years of his life. Though she may have learned of Claire's pregnancy, she thought her daughter died in the crash of Oceanic 815. She briefly saw Aaron at Christian's funeral, thinking him Kate's son. Years later, Kate approached her and told the truth. Carole took care of her grandson until Kate and Claire returned for him.
Since Then: Charlie, who'd long wanted to take care of a family, tried to help raise Aaron while on the island. Claire originally accepted this arrangement, and Charlie even helped retrieve Aaron after a kidnapping. Claire later pointed out that Charlie was not his father, and Charlie ended up kidnapping Aaron himself, convinced he had to baptize him. Before he died, Charlie left Aaron his heirloom ring. In his aferlife, Charlie again witnessed Aaron's birth and woke up soon after.
Origin: Claire found herself unexpectedly pregnant with Aaron after a relationship with Thomas.
Since Then: When Thomas broke up with her, Claire tried to give her baby up for adoption. A fake psychic warned her against this but later advised her to fly on Flight 815 to find a specific Los Angeles couple. Claire gave birth on the island. A mad woman kidnapped Aaron, and he later suffered from a fever that worried her. The Oceanic 6 escaped the island with Aaron, but Claire stayed behind and spent the next several years pining after her child. She eventually escaped the island, planning to raise Aaron with Kate. In her afterlife, giving birth to Aaron woke Claire up.
Origin: Jack saw examined Aaron's mother before his birth.
Since Then: Jack saw Aaron soon after he was born, and Claire occasionally consulted him about the baby's health. Off-island, Jack learned he was the child's uncle, and he initially refused to see him. He eventually changed his mind, and he helped raise Aaron. He read to the boy from the same book his father had read to him years before. He and Kate later broke up, and Kate said Aaron missed his presence.
Since Then: Kate occasionally helped with Aaron on the island, but she doubted she'd ever make a good mother herself. When she escaped the island though, she found herself carrying Aaron while Claire was left behind. She chose to raise the baby herself, claiming she'd given birth on the island. The guilt of raising someone else's child eventually convinced her to return to the island to find Claire again. In her afterlife, witnessing again Aaron's again woke Kate up.
Origin: Thanks to time travel, Sawyer watched Aaron's birth
Since Then: Sawyer never approached Aaron's mother, according to Hurley, but he eventually did, at Hurley's suggestion. They later discovered that Sawyer's voice kept Aaron from crying. When Claire abandoned Aaron and joined the Man in Black, Sawyer took the baby and later passed him to Kate, who eventually raised him. Sawyer presumably saw Aaron after escaping the island - his friends Kate and Claire planned to raise him together, and Aaron had become friends with Sawyer's daughter Clementine.