My wife is from Green and still has family in the area, so we're out there every other year or so.

In Cleveland, the Great Lakes Science Center on the lakefront or the aquarium in the Flats (but area isn't family friendly after dark).

Food wise, Swenson's drive-in is a solid local chain. Also recommend any family Italian restaurant you see, especially any that are multi-generational. My wife lamented that there were no good Italian restaurants in Utah when she came out to BYU 25 years ago.

If you are really into obscure church history sites, the lighthouse park in Fairport Harbor is an off the beaten path site. I've been told that the lighthouse then there inspired the hymn "Brightly Beams our Father's Mercy" (current lighthouse dates to 1871). It's also the primary port for the Saints arriving from northern New York when the church moved to Kirtland.