In addition to everything that's mentioned before (F5, hover over 3rd egg, memorize descriptions, etc), here's my technique:
I often double-click the eggs I want, because sometimes when I only click once, fast, it will have a little box around it as if I clicked it but it won't follow the link. Once it has a loading bar, leave it alone, don't click again. Right-click and open it in a new tab if you want, but leave the one loading.
Sometimes F5 is fastest. Sometimes clicking the "Cave" link is fastest. Sometimes hitting Tab and then Enter is fastest. I use a particular combination of these, usually, which I find to be pretty reliable.
If there's any egg that's not super-common in the cave (like, a Red or something), that's when I'll pay special attention. Once it goes back to pinks, balloons, mints, etc, then it slows down and people aren't going to click the new ones as much.
I've only gotten two rares in the AP: my gold (which was LUCKY - it was the only time I went to the AP that day, didn't even refresh. And it wasn't even a trade) and my first vamp.
But in the Cave, apart from the times I clicked once and it didn't load, or the times my scroll was locked, I've gotten just about every rare I've seen (which haven't been many, honestly).