I strongly recommend Manuel's place. It's a cosy and comfortable apartment placed in close vicinity of Barrio Carmen - a lively historical and nightlife area of Valencia. You can explore sights with ease from there, and if you get tired, relax at the terrace.
Manuel has been very friendly and accessible: our flight got delayed significantly, yet he welcomed us very warmly in the middle of the night. Great place to stay and we hope to come again.