The Cuyahoga River no longer catches on fire, but Great Lakes Brewing Company, which is located in Cleveland and has a restuarant and brewery tours, does have a Burning River Pale Ale.
Don't know where you are staying, but there is a House of Blues on Euclid Ave. Also on Euclid, a few blocks down, are some old restored historic theatres, they usually have traveling broadway shows. Check out Playhouse Square for more info.
Since you'll be drinking wine anyway, if you have access to a car, there are about 20 wineries located about an hour east of Cleveland on the Lake Erie shoreline. Turns out the soil and climate are perfect for growing grapes. Debonne and Ferrante are two of the biggest wineries. Just google Ohio wines.
Or you could get a cheap ticket to an Indians game, and go early and sit at the bar in the bleachers.
As others have said, there is Cedar Point, the Rock and Roll Hall of fame, and the Christmas Story house.
There are actually things to do there, you just have to look around.
Summer is a good time to visit, but it was the winters (and a job I hated) that made me leave.
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There's never enough time to do all the nothing you want.