I recommend Apartamentos Plaza Picasso 100%! Fantastic apartments, modern decor, clean, well equipped, helpful owners. Really, 10/10 quality and good value for your money. It's ideal for staying with family as there's a kitchen to prepare food and a huge living room with large windows. if I go back to Valencia, I'll be certain to stay there again.
I have no complaints about these apartments - they're wonderful, cheap and welcoming, right down to the last detail - utensils, washing machine, television ... they're in a very central area of the old town, so it's a bit of a trek to get to the City of Arts and Sciences, but you can walk through the Turia gardens. We walked everywhere, even to the beach, but there are bus and metro stops nearby if you prefer. The strangest thing is that you have to go elsewhere to collect the keys; on arriving at the apartments, we found a notice telling us where to go. We phoned and the receptionist explained how to get there - about a 10 minute walk, with our suitcases. If this situation were resolved, everything would be much better.