What should I look for, in a budget dropper post? It will go in a 2000s era hardtail, if that matters.
That old of a bike, I would assume it does not have internal routing. Also most likely it’s 27.2mm seatpost diameter (but not always).
So that is what you are looking for: a 27.2, external routed dropper.
Also measure seatpost extension from collar to rails, to make sure it fits.
chainreactioncycles often has good deals on their house brand posts
tranzX also sells direct to consumer now. And they are available from retailers. Note that they offer both air and coil versions, so make sure you are comparing prices in the same model.
I have used both those super cheap posts just fine.
next question is what remote you want. Do you run a front derailleur or did you convert the bike to 1x?
If 1x, get the cheapest under bar remote you can find that’s compatible with your post (most posts have the cable head in the post and pinch the cable at the remote, but not all).
If it’s still 2x or 3x, you will need to decide where and how to run the remote. Most are designed to fit on the left.
Personally, I used to run it on the right.
My thinking was that: if I was using the dropper, it meant the trail had switched from uphill to downhill or vice versa. In that case, I also needed to shift the front derailleur, so I wanted to do the dropper with my right hand.
You might be able to run an (included) under bar remote “upside down” on the right side.