In the Shinra Mansion, go to the safe on the second floor and use the combination 36-10-59-97. You'll have to fight Lost Number, but afterward you get some good rewards, including the Basement Key. Now in the basement you can get into the room with the coffins. Examine the one in the middle and Vince pops out. Keep talking to Vincent until you get to name him, then keep talking to him again until he just repeats the same line. When you try to leave the basement he will ask to come with you.