There was a drought when I was on vacation two weeks ago, and my tiny $10 meyer lemon didn't take it well. It was completely dry when I got home, leaves drooping. Within a few days all the leaves turned brown or fell off. The top half of the stem feels a little lighter and is becoming discolored. I assumed it was dead for sure but I just noticed a new leaf bud growing this morning. Should I cut the stem off above the new bud or just leave it alone and hope it survives? The stem up to the bud is still fairly green but it looks dying/dead near the top.

I would leave it alone for now--it may still re-green some of what I see. Once it has turned completely brown and you can see a clear border between the live and dead part you can cut it.

In treating citrus trees severely injured by flood, cold, mechanical, or drought injury , three courses are open: (1) leave them alone; (2) cut them back hard; (3) cut them back part way. Everything considered, Skeet's suggestion is correct, leaving them alone is the best plan. For some time following injury it is impossible to determine how much pruning should be undertaken. If cut back hard, more of the tops may be removed than necessary, and if headed back part way they may have to be pruned over again at a later date. There is no need for haste. Nothing can save trees, or parts of trees, that received to much damage. The exact extent of the injury cannot be determined until growth starts. It has been repeatedly demonstrated that there is a distinct relation between the size of the tree's trunk and branches and it resistance to injury. Small trees suffer much more severely that large ones. - Millet

After a few weeks there was an obvious line at that bud above which the trunk became brown an brittle. I cut it off at that point and it has been growing new leaf buds. It will probably be back to the pre-injury size in a month or two. I thought it was a goner.

I am glad to hear it is recovering--I knew it would.