The morgue employee worked at the morgue where Edmund Burke's body was held after his death. Juliet visited the body and was told by the medical examiner that she had to sign some papers, but that she could take her time. He then left her alone, where she met Richard Alpert and Ethan Rom.
The character had two cut lines, which were simply the morgue employee confirming that the body was Edmund's, and then confirming that Juliet was Edmund's ex-wife. (The Lostpedia Interview:Steven Labrash)
Although being an unnamed character, the morgue employee is in fact named, as his name is shown on his staff badge. However, his name is unidentifiable in the episode. Therefore, he is considered as an "unnamed character".