Adair Trevelyan picked Champion because he wanted to be a protector. And because it's good to be a Champion. His backstory was that he ran away from home to skip templar duties, so that was a no. But he was still very Andrastian so anything that sounded like blood magic was also no.
Lavone Trevelyan picked Rift mage because she felt it understanding the breach magic would help her deal with it and to use her mark more efficiently. And she didn't want to be commanded by a knife ear nor fiddle with corpses.
Baldur Cadash waffled too long on the issue and ended up not specializing at all. But he would have chosen Champion, since he liked to turtle up and Champion sounded like a noble life style. And he was a sucker for being "Blackwall lite" minus darkspawn.
Bergstein Cadash picked Assassin since I figured he had already specialized in that before the Inquisition and wanted to refine his skill even more.
Cadogan Trevelyan picked templar because I head canoned he had already started taking Lyrium in the order, but was still just getting it in rations. So now he could dose himself while not letting his training go to waste. And smiting evil felt so good.
Chayim Adaar picked Champion because of out of three evils that seemed best. And he wanted to stay on his feet for longer periods of time in battle.
Cressida Cadash picked Artificer because she loved toys that exploded on impact. And she wasn't a good fighter by herself since her job in the Carta was selling stuff and public relations, so the items also balanced that out.
Isalath Lavellan picked Reaver since her fighting style was already pretty savage. Ripping into her foes and even biting them. So eating them was the logical next step. And eventually turning into a dragon sounded hot.
Kaliesa Adaar picked Artificer for a similar reason as Cressida. She started training her skill late in her life, so she needed to tip the balance with some tools. She also relied heavily on poison.
Mytjan Lavellan picked Tempest because then he could light himself on fire. And that sounded awesome. Assassin also sounded to grim for him and he could never focus long enough to create any kind of traps or bombs.
Ramona Trevelyan did not specialize in anything, feeling that she could just get by on raw skill. But she did slightly fancy either Knight Enchanter or Rift Mage.
Saerin Lavellan picked Rift magic since it bolstered her own magic and having base understanding in the magic Solas used would make it easier for her to squeeze some secrets out of him.
Vedis Lavellan became a Templar because she felt powerless against the magic thrown at her and wanted to be able to protect her team from it. And she was to daft to realize the damage Lyrium might cause her later in life.