Around the Plaza de Oriente and Calle de Vergara for the full royal-quarter effect; Calle del Arenal for the direct line to Sol; toward Plaza de España for newer stock and the gardens. Evenings are quiet everywhere here — this is the centre's best barrio for actual sleep.
Palacio & Ópera lists around 230 bookable hotels and guesthouses, from roughly €46/night. Prices on the area page are live; booking 3–6 weeks out usually lands the best rate, with free cancellation on most rooms.
If it books out or you want to compare: Sol (from €45/night), La Latina (from €40/night), Gran Vía (from €48/night) — each with its own area guide on this site.
Curated picks are coming — meanwhile, the live search covers every bookable property at the same price or better.