Why didn't Lily or James become their own Secret Keeper?

Answer 1

It was supposed to be Sirius Black but they changed at the last second - without telling anybody - because Sirius thought he would be the obvious choice. They then changed to Peter Pettigrew, who was of course a traitor.

Answer 2

Lily and James Potter went into hiding with their young son Harry because Albus Dumbledore told them to.

Lord Voldemort heard a prophecy (from Sybill Trelawney, but heard his man Severus Snape) that named Harry Potter as Voldemort's equal enemy. Although Dumbledore didn't tell Lily and James, Voldemort wanted to kill Harry.

Dumbledore put a Fidelius Charm (that will hide a house from anybody who seeks it unless the Secret Keeper [one person entrusted with the secret of the charm] confides it) on the Potter house in Godrics Hollow, but the Secret Keeper Peter Pettigrew, the traitor, told Voldemort of Lily and James's whereabouts. You know how the story goes from here :)

Hope this helped.

Answer 3

Because the people who have the Fidelius Charm on them, in this case, Lily and James Potter, have to put their trust in someone else they know who won't tell their secret.