I don't know what it is, but I always get choked up when Valjean sings "...tell me my child what is your name?" and the solo woodwind that follows, ending in Cosette telling him her name.
I guess because it's such a defining moment and the way it's set to music makes the eyes misty.
Depending on who's playing Valjean, sometimes the actor does something that adds to my already choked up reaction; after Cosette tells Valjean her name, I love it when he whispers partially to himself "Cosette!" because it sends the message he is relieved to have found her and was likely out looking for some time. And she has lost her mother and doesn't know it, so when Valjean is relieved, you can sort of relate since we've been rallying for Cosette the whole time and want Valjean to find her and rescue her from the evil Thenardiers.
There's many great moments in the show but, for some reason, that one gets to me more than most.
Recreation of original John Cameron orchestration to "On My Own" by yours truly. Click player below to hear.