It’s would have. As in, I would have written this earlier if it would have dawned on me sooner. Not the would have thing. That’s just commonsense. Have an auxiliary verb meaning “to be required” or “to be compelled.” This is opposed to would of which doesn’t make a lick of sense. (You might say to me, “Dawson, lots of English doesn’t make any sense.” You’d be right, but the reason English doesn’t always make sense is because people like you have been misusing it for centuries.) Of is a preposition. It indicates relationship. No, the thing that dawned on me was that people write would of because they mistake what they are saying. They’re saying (whether they realize it or not) would’ve.
They do sound rather alike, but I’m thinking that this whole mix up started with the word wouldn’t. You see, wouldn’t is a contraction and people have forgotten that you can (and we do!) contract more than two words together. At some point people started saying wouldn’t’ve instead of would not have — which is a horribly unwieldy phrase. When they wanted to write it out they wrote wouldn’t of because all those apostrophes just look so weird next to each other. From there it was a short step to would of.
So stop it! You and I are going to start writing double contractions until they’re popular again. We’re going to fix the language. Let’s shouldn’t’ve, wouldn’t’ve, couldn’t’ve our way back on track.