For boats that don't require titles (such as kayaks), "proof of ownership" can be as simple as filling out a form that basically says "I own this" (includes lots of boat info including hull ID number, length, hull material...) and you sign it and get it notorized. Easier method is if you have the MSO. Go through any paperwork you got with the kayak and look for the MSO. It often has some sort design on the paper it's printed on... a bit similar to a vehicle title... it will have the hull ID number, the length, material and so on. If you can't find it, don't sweat it, you can always fill out the ownership form and get it notorized.
Bubbagon and fishwendel2 mentioned the alternative registration. This applies to all "hand-powered" watercraft... it costs $25 (instead of $20) for 3 years, but you don't have to put the "OH" numbers on the sides of your boat... it's just a single registration sticker, with a few options of where to put it. If you go with the traditional registration, you much display the boat numbers AND a reg sticker on BOTH sides of the boat. With the alternative registration, you are never allowed to attach a motor to the boat (though few kayakers do, but it happens... you couldn't use Ohio's alternative registration for the Ocean Kayak Torque).