April Ryan in TLJ is my favourite game character of all time... also the game has the most bitter-sweet ending I've ever experienced - both climactic & anti-climactic.
Kate Walker in Syberia, the way she changes throughout the journey culminating in the 'choice' she makes at the end... and also the ending of Syberia 2. A moment involving her & Oscar (the automaton-butler) that was supposed to be emotional didn't do anything to me, though.
And heck, Bobbin Threadbare meeting his mother in Loom made me cry when I was an 8-year old kid playing it as one of my first video games ever.
EDIT: not to forget, a certain moment in Beyond good & evil involving a ruined, empty lighthouse, Jade & her dog Woof.