Chibi-Robo. Not sure if you mean metaphorical or literal basement, or whether or not the character in question has to be the owner of the basement ('cause in Chibi-Robo the family who has Chibi owns the basement), but yeah, what you find in the basement of the house is a big plot point of that game. (I feel like you probably might've played Chibi Robo, but just adding my suggestion).