Of the few games I've played in my life, I think Earthbound is probably in the top 5 of games that made me really emotional (Read: Not Cry) but got me close. I think when I hear "Smiles and Tears" at the very end, I can't even listen to the song anymore because it gets me on the verge of tears.
My Top 5 Video Games that make me cry:
5. The Last Guardian video/ A video that put the audio from Where The Wild Things Are Movie trailer with the game, makes me cry doubly hard.
4. The Thought of Killing Your Girlfriend Jennifer in Splatterhouse, and the unsettling sound her shouting "Help Me!" as she transforms. The Turbographx 16 version, not the remake I haven't played.
3. Chrono Trigger for the SNES (or DS, if you prefer) and "Memories of Green". Just listen to it. Think about something that broke you up, you'll bawl, guaranteed.
2. For me, personally, the thought of killing a little sister in Bioshock, which is why I probably won't play the game more than once.
1. Earthbound, perhaps the entirety of the final battle, as well as the last song in the game. I have heard Mother 3 (Or Earthbound 2) can do it too, but I can't find a copy, and I legitimately want to buy it, maybe that will make me cry.