It took me a while to find the jewelry findings that most met my requirements. Of the several ways to hang a pendant on a chain, the one I use most often is the bail. I like the Aanraku bail because it looks good, is available in 3 sizes and is avainable in Sterling plate, gold plate and nickle plate. It has plenty of surface area for the adhesive to secure it to the cabochon. They are a staple in my supply caddy.
I also use Aanraku bails for my dangle earrings, though sometimes if I need something smaller (or less expensive), or if I've run out of Aanraku earring bails, I will use beaver tails. They are simple, inexpensive and very effective. I just bend the opening forward so it sits more centered on the earring, allowing the earring to hang straight.
I've tried several different clasps. I used to use the barrel clasp most of the time. But because they sometimes work loose, I've settled on the hook and eye clasp. It's simple, inexpensive, easy to secure, and unlikely to accidently disconnect. The one advantage the barrel clasps have over all these other types is you do not need a jump ring to secure them. The ends of the barrel clasp function as the jump ring and all you have to do is loop it through your chain with some small pliers.