Repairing a leaky valve stem
I think I have a few solutions. One, I could use graphite packing string on the stem threads. Two, I could use teflon tape on the stem threads. Or three, I could use plumber's grease on the stem threads.


None of those proposed solutions will work because the problem has nothing to do with the "stem threads". Your problem lies with the missing packing. The valve design incorporates what is called a packing gland, which includes a packing nut, a stuffing box, and packing material.

The stuffing box is where packing material (in your case probably a washer) is supposed to be. The packing nut screws down onto the packing, causing it to squeeze around the stem, which prevents the water from passing through.

You should take the valve stem to a good plumbing shop, where they will have a wide selection of various size packing washers to choose from. If none are exactly the right size, it is very common to trim down a larger packing washer with a knife, to make it fit. It is also very common to use packing string (you mentioned it in your post) in lieu of, or in combination with, a packing washer.

This pic illustrates the theory but instead of a packing yoke with bolts, your valve uses a packing nut: