There are two options I can recommend:
1) tel. +385 (0) 98 647 443, +385 (0) 20 422 840 - Andrija Glavinic is the guy we stayed with. He is nice and inexpensive. Allthough I am not sure that prices would stay the same If you are foreigner. So if you call him, tel him that you have an Croatian friends Nina&Davor who recommended him because of good prices. It is not in the middle of the city, but not so bad.
2) seccond option is just to go there and you will see many places with signs "rooms" and "zimmer frei" and just negotiate your stay on the spot.
Renting ahead with the usual way is usually expensive and also you could book something which is not close to old city. One very important thing: having a car in Duborvnik can be very messy and expensive! There are huge praking and space problems, so it is very difficult and expensive to leave a car. I am not sure about your itenerary but maybe you can return the car in Dubrovnik or leave it somwhere outside the city and use public transportation. IT is not so bad, although not very cheap. If you call Andrija or anybody else, ask him for a parking spot.