Apparently, Star Trek: The Next Generation's counselor Deanna Troi has a space-Israeli accent. However, her mother, Lwaxana Troi, has a space-American accent.
Why, then does Troi have a space-Israeli accent? Is it something to do with the accent her father has or is there another in-universe reason for it?
Troi's accent is typically Betazed, at least according to an interview with the actress portraying Deanna Troi, Marina Sirtis;
They said "OK, you've got the job. You have to do an accent". And I said "Well, where do you want the accent to come from? and they said "Betazed" [audience laughs] So I though "Great! I can make up an accent. I mean, who's going to tell me that I'm doing it wrong? Hands up all the Betazoids in the audience..." [Marina is the only one who raises her hand. Audience laughs again] ...so I made up this accent".
As to why her accent is rather different from her mother's accent, the show's producer originally explained to her accent was similar to her father's. After TNG: Dark Page was produced (in which her American-accented father appears), Rick Berman then explained to her that it was because she had been sent away to boarding school.
Now I assumed that I was doing the Betazoid accent. But then mum came along. Well obviously she was from the American sector on Betazed [Audience laughs]... because she didn't have an accent. She had an American accent. So being, as I said, one to make trouble I go to Rick Berman's office and I knock on his door and say "excuse me, I thought I was doing the Betazoid accent but obviously I'm not because Majel is American" and of course by this time we're well into the filming and they're rewriting scripts constantly. They're really busy. They're casting, they're editing, they're doing all this stuff and the last thing they want is Marina Sirtis knocking on their door and being a pain in the butt. So Rick Berman, and I know all he wants to say to me is "shut up and get out of my office", but he has to come up with an answer so he says, "you know what, it was your father's accent". And I said "Really? Because I don't believe that because of my relationship, Deanna's relationship with Lwaxana was kind of identical to my own relationship with my mother, God rest her soul. And I knew that they had lived together in the same house. That was why there was so much friction on the show". So he was like "No, it's your father's accent and go away".
Well then I think it was Season 6 or Season 7, I don't remember because they've all kind of blended together at this point, Dark Page, where we met my dad [she pauses for effect] who was from Iowa [audience laughs]. And I just had to. I know we were coming to the end of the series and I had to go back to Rick Berman's office and ask. So I knocked on his door and I said "You told me back in season 1 that I was doing my father's accent but actually we've met dad now and he doesn't have an accent" and Rick is ready to throttle me. He's like "Really, this is the conversation that we're having now?" And he goes "They sent you away to school! Now shut up and get the hell out of my office!!
Troi's accent is not typically Betazoid based on evidence. Other Betazoids, notably Lwaxana Troi, Devinoni Ral, Tam Elbrun, and Lon Suder, don't.
There is never any explanation given for the origin of her accent in universe. As you you've pointed out, Deanna's mother, Lwaxana doesn't speak with such and accent. In addition, neither does her father, Ian Andrew Troi, depicted in "Dark Page" around the 25:00 mark.
The bottom line is, there's no in-universe explanation.
Out-of-universe has been covered in the linked question, but with respect to the backstory out-of-universe, one presumes that at the time that Sirtis and the producers decided for her to use the accent she does, the mindset would have been that it was how Betazoids talk. Later, when Lwaxana was added, it was decided that Majel Barrett couldn't or didn't want to play the same accent. From there on, no other Betazoids ever played the accent either. And the last holdout explanation would have been her father. But when he was depicted in "Dark Page" it was decided again (if it was even considered) not to play the accent. This would have been a strange justification anyhow, since he is a human. (Of course this entire paragraph is mere speculation.)
