At first,I believed that Troi's accent was meant to be Betazoid accent.
It was. At one point, Betazoids were going to have either elaborate forehead makeup (but they preferred to avoid putting beautiful women under too much forehead stuff - ie. Terry Farrell as Jadzia Dax did one screen test in a latex forehead!) or, believe it or not,
three breasts. DC Fontana asked Roddenberry if the breasts would be horizontal across Troi's chest or vertical. According to an interview I did with Bjo Trimble, before "Encounter at Farpoint" aired, the producers had decided "to portray Troi's alienness through gesture and voice".
At the point where Sirtis developed her unique Betazoid accent, based a little on her own Greek upbringing, there was
no idea who would be playing her parents one day. It was Majel Barrett's idea to play Mrs Troi as "Aunty Mame", complete with brash American voice. At conventions, over the next few years, the production staff started to suggest, when asked, that Troi's perhaps-Greek accent (and surname) "probably came from her human father's side". At that point, no one knew that they'd ever write or cast "Dark Page". He turned out to have neither Deanna's accent nor Sirstis's Greek heritage.
I think one of the novels speculated that Mr Troi had followed Betazoid tradition and taken his Betazoid surname from Lwaxana when they were married. (Hence, also, the "Mr Troi" line to Will Riker in "Nemesis".)
Remember, also, the original concept for Troi was a willowy blonde. That's why Denise Crosby was screen tested. Away Team leader, the compact, brunette Macha Hernandez, the original template for Yar, was originally described as a "Jeanette Goldstein-type from
Aliens", and Marina Sirtis was the top contender.