In the fourth episode of Star Wars, A New Hope, Obi-Wan tells Luke that he hasn't used "Obi-Wan" since a long time before Luke was born.
But in the third movie, Revenge of the Sith, Obi-Wan uses his name through the whole movie. Is this an error in the third movie, or can we just assume that Obi-Wan has a bad memory?
As you can see here, the Ben alias is a way for Obi-Wan to stay incognito on Tatooine. He moved there after the events of Revenge of the Sith, 19 years before A New Hope. We can safely deduce that Obi-Wan assumed the alternate identity of Ben as soon as he relocated, and those 19 years can be thought of as "a long time".
Obi-Wan didn't tell Luke he "hasn't used 'Obi-Wan' since a long time before Luke was born." Here is the actual quotation, directly from the script of A New Hope:
BEN: Obi-Wan Kenobi...Obi-Wan? Now thats a name I haven't heard in a long time...a long time.
LUKE: I think my uncle knew him. He said he was dead.
BEN: Oh, he's not dead, not...not yet.
LUKE: You know him!
BEN: Well of course, of course I know him. He's me! I haven't gone by the name Obi-Wan since oh, before you were born.
So he said he wasn't called Obi-Wan for a long time, and this time was roughly before Luke was born. In fact, I checked the Revenge of the Sith script and he had been called Obi-Wan by Padme just after, but Yoda also called him "Master Kenobi". In any case, "before you were born" is a decent approximation for events that occurred 19 years prior.
Could it be possible that he doesn't "have" a name at all in those wilderness years of hermitting? He wanders in and out of town only rarely, and doesn't introduce himself to anyone. "Old Ben" is just a chinese whispers of "Obi Wan"- reflecting Owen's lack of understanding of what the real score is perhaps. He's not gone by the name of Obi wan, but he hasn't been telling everyone his name is Ben either. When Luke sees him, and refers to him as Ben, Obi Wan just smiles, he doesn't say "yes, it's me- Ben" or any such nonsense.
Obi-Wan uses the name Ben in season 4, episode 15-16 of The Clone Wars I still dont know why he calls himself Ben from time to time but its more than once, some kind of alias.
There's an additional explanation to Obi changing his name to Ben, apart from stay incognito, so far I know Satine used to call him Ben in privacy and he admitted he like that name
The reason on why Obi Wan calls himself Ben is something that can't be answered in only one way, especially now that Disney placed a ton of stories out of the way.
I researched all the possible reasons and compiled them in this star wars post.
In Star Wars, the movies, he calls himself Ben for those reasons, during the Clone Wars he calls himself Ben because it is his code name for infiltration missions.
