When the Jedi Order was wiped out thanks to Order 66 it was best to go by a name that wasn’t known to the Empire, so Obi-Wan Kenobi became Ben Kenobi, which isn’t such a big name change really, but it was good enough for Tatooine since this is were Obi-Wan set up in order to watch over Luke. Some might wonder just why the Emperor and Darth Vader might have never looked upon the planet for either Kenobi or Vader’s son, but the best explanation is that Vader was told by Palpatine that Padme, his wife, had died in childbirth, and it was assumed that the child, or children, had died as well. It does feel as though the explanation was killed off prematurely since Anakin was likely strong enough to feel another Force user on Tatooine, and hew as likely strong enough when he was still a Jedi to determine that his wife was having twins. However, it does feel pertinent to mention that his mind was probably a bit clouded given that he was dealing with a lot of stuff at that time, such as fighting a war and trying to decide between his loyalties. Has anyone ever tried to concentrate on one thing when a dozen different things demand your attention? It’s not easy, and it can tear your focus to shreds.
Back to Kenobi though, it’s also likely that his name might have been added to a hit list of sorts that the Empire could have had on file, a way to eliminate any rogue Jedi that was still at large in the galaxy. It’s not likely since Palpatine took great care to send Vader and his inquisitors and other agents after any possible Jedi to take them down. But it does sound like something the Empire might have done at one point just to ensure that the threat was completely gone. The Moffs wouldn’t have cared, as they were more about quibbling to see who curried more favor with the Emperor as they tried to one-up each other and elevate themselves over each other. Only one of them ever had Palpatine’s ear in a way that gave him any leverage over Vader though, and that was Grand Moff Tarkin. But as far as Obi-Wan went, even Tarkin believed that he was well and gone. On Tatooine, Kenobi could become Ben and hide away from the rest of the galaxy in order to watch over Luke and keep him from danger as much as possible. It does still feel that Vader and Palpatine might have been able to ferret Kenobi out had they taken the time and effort, but since Ben didn’t make himself that much of a nuisance to anyone he slipped beneath their notice, as they were set on the overall conquest of the galaxy.