Airbnb is a great option, just look for Montrose/Hyde Park area.
Houston Heights (around 19th Street) and Rice Village are also very nice neighborhoods with cool places to eat nearby.
Hotel
Best Western Plus Downtown Inn & Suites
915 West Dallas Street Houston, TX 77019 (713) 571-7733
Best Value option in Montrose. You can easily walk to the Bayou from here
Hotel
Hotel ZaZa Museum District
5701 Main street Houston, TX 77005 (713) 526-1991
Beautiful historic hotel. Next to the Natural History Museum and Hermann Park
Hotel
Aloft by Marriott Houston Downtown
820 Fannin Street Houston, TX 77002 (713) 225-0200
Hotel
Club Quarters Hotel Downton, Houston
720 Fannin Street Houston, TX 77002 (713) 224-6400