I stayed at the Hotel Goya which is basically in the Barrio de Santa Cruz. Loved the location and the hotel, which was a basic budget hotel but great. I think it gets good reviews on TripAdvisor and might have been suggested by Rick Steves' also. If you desire more luxury, there are many others in that area, but I highly recommend the area in or near Santa Cruz. The Giralda, the cathedral and the Alcazar are also right there, less than a minute walk.
The Alameda de Hercules, suggested above is a bit more out of the way. I've visited there, but it's a bit of a walk from the sights. Also I much prefer the atmosphere on the area of Santa Cruz, closer to the sights. You'll love the restaurants and cafes in the little calles and plazitas. Just take a tour using the Street View feature of Google Maps.