Cleveland is fun, but I've never done it with little ones. We like the Rock & Roll Hall of Fame and going to the ballgame (one of our shared goals in life is to catch a game at every MLB park). Cedar Point isn't far from Cleveland and that's a great time but again, not so much for little ones. There are some nice indoor waterpark resorts in the area, but I think of that more as a winter escape rather than a summer vacation.
In Ohio, Columbus is my favorite. Great science museum and zoo, and not too expensive because it isn't much of a tourist destination. Toledo is fun for a weekend because they have an awesome zoo, but the science center there closed due to funding issues so there's not much else for little ones to enjoy.
Have you ever driven up the west coast of Michigan? The Lake Michigan coast has a lot of fun outdoorsy stuff to do, and we've done trips up north with kids of all ages because my family has vacation property just inland from the sand dunes. There's an amusement park in Muskegon and a couple of waterparks along the coast, and in the Traverse City area there are a ton of family-focused resorts. On the other side of the state, the Detroit/Ann Arbor area is fun too - Henry Ford/Greenfield Village (
The Henry Ford: America's Greatest History Attraction) are great for little ones who are into cars and trains and animals, the zoo is pretty cool, and the Detroit science center is a lot of fun. In Ann Arbor, there is a museum that is all hands-on that was a hit with my kids even as toddlers (
Ann Arbor Hands-On Museum).
Another thought might be the Wisconsin Dells. It is *very* touristy, but I really liked the indoor waterpark resorts when my kids were very little because they are small-ish, contained, and easier to keep the kids in safe water than in lakes/rivers but with lots to do.