It's an informal response meaning "I don't really mind". "Do you want to go to an Italian or a Mexican restaurant?" "I'm not that bothered, to be honest. You can choose."

Hi, you're right, although "I don't care" is a bit stronger

Sorry, I just saw your actual question don't take the advice from my previous answer. I was a little confused as to what the question was. "I'm not bothered" literally means what you have said. However, like some of the other posts on here, I haven't heard it in the context that you provided. There is the possibility that the person who said "I am not bothered" was talking about something in a particular restaurant that doesn't bother them. That would be my best guess or it could be the difference between English dialects. I hope that this helps.

Native U.S. English speaker here. People say, "I am not that bothered BY IT/SOMETHING," because something has to not be bothering you. People might be more likely to say, "It doesn't bother me". It means: I'm not upset. I don't care about it. It has no effect on me. I have never heard anyone say "I am not that bothered" in the context of talking about what restaurant to go to. Maybe it's a British thing? I don't know, but no one says that here in that context. People say, "I don't care," or "it is all the same to me" when talking about where to eat, but not "I am not that bothered."

If you are talking about the context then it specifically means that a person is recommending a restaurant possibly to another person. I hope this helps. Let me know if you have more questions!