Ohio, to me, has always been that state you drive through to get somewhere else (sorry Ohioans!), and I almost had the same experience with their digital archives. You have to look carefully to notice what's there, but once you do, there's a good deal to appreciate.
The Ohio Historical Society Archives houses the official archives so there's a big focus on government materials, along with assorted other online collections. Their "Digitization Projects" include:
There's also the multimedia collection (mostly photos) of the Joe Munroe Archives, a photojournalist, film maker, and environmental activist
Family history types will appreciate the Ohio Online Death Index, 1913-1937...the index information will allow you to order a copy of the death certificate.
Over at Ohio Memory you can browse or search a substantial collection of the usual: postcards, posters, brochures, photos and all sorts of other ephemera that make up a state's history. Good luck with the subject directory, which is a large unwieldy list, but you can also search for specific items or take advantage of some of subsets of materials:
or any of the following Select Topics, each with a link to a summary write-up (PDF) along with actual archive materials:
Two other tidbits to check out from other sources:
Newspaper archives, covering everything from the Akron Daily Democrat, and the Anti-Slavery Bugle, to the Xenia Sentinel
The Cincinnati Library has a nice collection of old city directories to view or download: