For longer stays, try the Gym Club Suites [520-234-5921], also in "downtown" Bisbee. This building was converted from the YMCA to a group of suites with full kitchens.
The Copper Queen Inn offers more expensive accommodations. Used by the mining administration during Bisbee's hey day the Queen is probably the the largest of the hotels in the Old Bisbee area.
Numerous other accommodations are available in the the town up and down main street, or try Shady Grove just down from the pit. They offer a mixture of vintage trailer homes and a yacht as sleeping quarters.