Hello. I just purchased some inexpensive 1/16 oz drop shot weights that I didn't realize are lacking the pinched eyelet that traditional drop shot weights have. What knot should I be tying to the eyelet that should break if the weight gets hung so I don't lose the entire rig?

I just use a double overhand knot figuring that has to be plenty weak to break off at the weight instead of the hook or anywhere else in the line.

Could use a 3 wrap uni knot, or a double overhand knot. I've used both, you really don't need a super strong knot for that

Arbor knot

1/16th is awfully light.  I don't hardly use anything less than 1/4th oz.  3/8th is my most often used size.  

I tie a perfection loop for all my sinker on the bottom rigs.

Another vote for the double overhand knot.  It'll break long before the knot holding the hook.

Not quite as fast as the overhand but I use a 3 turn clinch, hold the improvement.

Just an overhand knot for me.  

 

Also, I have found, depending on the type of weight, if I tie two on the same line I get hung up less.