TIL the Ohio History Center in Columbus is free on Sundays, and my wife and I turned it into a whole afternoon
I figured it was only for members, but the front desk person at the Columbus museum told me last weekend that anyone can walk in free every Sunday, and we ended up wandering the 1950s street exhibit and sharing a $6 pretzel in the cafe for the cost of parking, so now I'm hunting for other places with free days, and if you know any in central Ohio I'd love to hear them.
The Metro Parks are ALWAYS free, that's not a free day thing. But the Franklin Park Conservatory does free Sundays for Columbus residents, that one's actually worth it.