Yeah, the East Red line is pretty much a bust, running through rough neighborhoods with nothing resembling a hotel anywhere near it. (At the end, at the Windermere station, you can catch a Laketran bus out to Lake County, but I do not know the laketran service area or what motels/hotels it might visit.
The Green line is pretty much all residential.
Aside from staying AT the airport, I am not sure what sort of lodging you can find along the West Red/Airport line. (A westsider might drop by to give better info on that line.)
The Blue line might be an option, with a small hitch. It ends where Van Aken Ave. runs into the intersection of Warrensville Rd., Northfield Rd., and Chagrin Blvd. and I cannot think of any motels near that intersection (that don’t rent by the hour). However, two miles East, in beautiful Beachwood OH (and adjacent but not quite as beautiful Windermere, OH), there are a number of motels/hotels on Chagrin Blvd, Park East Dr. (Beachwood) and Orange Place (Windermere) (all use zip code 44122, I believe).
Bus Route 5 (Weekdays), (Saturdays), and (Sundays) passes through that area, with a special loop to pick up hotel patrons from Park East. (Orange Place folks may have to hike the half mile down to Chagrin Blvd.–I really don’t know.)
Here is the Cleveland RTA index page. From there you should be able to find maps (.pdf) of the whole system, (bus and light rail), prices, and ways to transfer. It would probably be a good idea to check back before your trip to see what might have changed–although I would not expect them to cut the hotel loop buses in the summer.
Park East is the first street extending South from Chagrin Boulevard just west of the Chagrin Exit for I-271. Orange Place is the first street extending South from Chagrin Blvd just East of the I-271/Chagrin exit.
Here is a list (if the link works) from Yahoo! Yellow Pages of the Hotels in that neighborhood (after clicking past all the advertising pages that did not provide addresses):
Beachwood hotels
The same list copied from the page in case the link failed:
Clarion Beachwood 26300 Chagrin Blvd (216) 831-5150
Cleveland E Beachwood 3663 Park East Dr (216) 464-5950
Courtyard-Cleveland Beachwood 3695 Orange Pl (216) 765-1900
Embassy Suites Hotel 3775 Park East (216) 765-8066
Extended Stay America 3820 Orange Pl (216) 595-9551
Hampton Inn-Beachwood 3840 Orange Pl (216) 831-3735
Hilton Cleveland East 3663 Park East Dr (216) 464-0871
Holiday Inn Cleveland-Bchwood 3750 Orange Pl (216) 831-3300
Homestead Studio Suites Hotel 3625 Orange Pl (216) 896-5555
Radisson Hotel 26300 Chagrin Blvd (216) 360-7341
Residence Inn-Beachwood 3628 Park East Dr (216) 831-3030
Super 8 Motel 3795 Orange Pl (216) 831-7200
There is also a new Marriott a mile South on Harvard Rd at Richmond Rd, just West of the I-271/Harvard exit, but I did not look to see what busses might service it.