JimC once bubbled...
This first knot will hold and is easy to tie.
The second is the the water knot and will hold. The grape vine knot is more secure, but requires an extra wrap around.
You don't need bomb proof, just tight. Not like your life is hanging off this 1/8" bungy. 
I use the second knot pictured because it has a nice flat profile and wonl't slip or undo when theres no load on it.
Hate to nitpick, but the first is a square knot (which is fine for this use), and the second is not a water knot. An actual water knot is more or less ideal.. it's small and very strong. It's typically used to tie ends of webbing together.
A water knot is where an overhand knot is tied in the end of a rope, and another end is thread backwards through the entire knot. It's resistant to coming undone, but typically used to tie webbing together since it allows the webbing to sit flat against itself. It might also be called a tape knot.
The second is most commonly referred to as the fisherman's knot (although the term fisherman's knot generally refers to something else entirely). This is a decent way to tie two ends of a rope together, but it's fairly low strength. The double fisherman's, which involves an extra wrap on both sides, is very strong, and the ideal way to join two ropes (be sure you backup the knots).
Enough of that.. I use my bungee for something else entirely, but I use a water knot when I want to fasten a loop, and a fisherman's for my bungeed safe second. The fisherman's is a double slip knot, so i can make a nice big loop, shove the reg mouthpiece in it, then just pull the ends and tighten it. I think this is the fastening method DIR advises, but I'm not sure.
