If you're looking to just put a stopper knot on an end of line, a simple overhand knot will do.
If you want a knot that can easily become untied, a figure 8 knot would be better.
If you want a bigger knot that won't come undone, do an Ashley's Stopper knot.
No matter what knot you choose to use, whether it's fishermands for a reg necklace, or your regular overhand for a permanently attached necklace, or even something completely different of what ever, here's what you should do.
Before you cut your bungee to length, prep it by wrapping masking or painters tape around your bungee and cut in the middle of that tape. Next, take a lighter and melt the end. The tape should stay intact and your bungee line will have a nice melted end, without your outer sheath banana peeling away from your inner cord. Remove the tape and you're now ready to tie your not.
After you prep your line, tie your knot (pulling it tight with pliers helps a lot) and lightly heat the knot to stiffen the bungee. Be careful not to burn the outer sheath.