Maybe it would have been better to say that their accounts are more believable than if there was no mention anywhere of a potential source of the information.
That's true, but we know the bare bones of the account are correct because Josephus mentions the execution of John the Baptist.
There's no potential source, by the way, for a lot of things that are mentioned in the gospels. There's no chance that anybody knew what really happened when Jesus was alone in the Garden of Gethsemene, nor at his trial before Pilate or even any of the events after the disciples all fled following Jesus' arrest.
They do indirectly: there were women present at Jesus' death who had followed and ministered to him in Galilee, according to both.
But she is not named, so it is pure supposition on your part to assume she was there.
I recall from GCSE Religious Studies that Luke relied quite heavily on the testimony of women. Would he have chosen to make this up, assuming that a man's testimony might have been considered more reliable at the time?
Luke relied mostly on Mark, Matthew/Q and the Old Testament. There's no evidence in Luke of him using any testimony from anybody who was there. In fact the prologue seems to deny it, claiming he used earlier written accounts.
