Accommodations in Dubrovnik

Budget?

I stayed at and really liked Villa Dubrovnik.

I liked Apartments Pavisa, recommended in the RS Croatia and Slovenia book. Near Buza Gate inside the walls. It's a bit of a climb but convenient location.

We stayed at an AirBNB. Amazing

I stayed in an apartment called Roko House inside the city walls, but I would not hesitate to also stay outside (if your budget warrants it), as transport is really easy in Dubrovnik. I loved the apartment (although small, it was very bright, light colored, and clean) and the lady who managed it was lovely.
https://www.hotels.com/ho367323/roko-house-dubrovnik-croatia/
http://www.rokoapartments.com/?file=about-us

This link also has a lot of options and is fun to look at. There are some fabulous and interesting properties out there.
http://www.dubrovnikapartmentsource.com/dubrovnik-luxury-apartments

If you want to save money and avoid hordes of tourists, stay outside of Dubrovnik. We stayed across the bay in Cavat at a beautiful apartment with sea views for next to nothing, and took the 10 minute boat ride into Dubrovnik (took the bus home because the boat stops running in the early evening).