I'm a huge fan of vintage banjo ukes and have five.
What to look for? All metal parts present and in good shape has been mentioned, as has the neck, but I can't stress either enough. Also - for cleaning metal, instead of using a cloth to polish, use 0001 steel wool. Works much better and can also be used to flatten imperfections in a skin head.
You want to also make sure that the laminate on the pot is intact and not cracked. This won't effect playability, but it sure signals that is wasn't well taken care of. It can be fixed, though won't ever look quite right.
Finally, you want to make sure that the pot has no structural issues. Often, models with thin pots, like the Stella or Oscar Schmitt, can go out of circular over time and under normal pressure (they honestly weren't top quality ukes, so no surprises there). Thicker pot models can sometimes be separating out of sight under the head or resonator. You want to make sure that's not happening - and a seller can take the resonator off to make sure that end is ok, though often, with eBay, I don't think sellers even know that heads can come off and go back together quick as a wink and they aren't willing to try it. That's something to ask the seller about, but definitely check on when you get it.
Lastly, keep in mind that there are roughly three levels of vintage banjo uke out there - cheapies - mid-level - and professional grade. Every flavor of Gretsch is a cheapie, every flavor of Gibson except the UB-1 - is a professional.
If you go with a cheapie - cost between $50 and $200 in very good to excellent condition - you'll want to make sure you have good wood and all the metal in place. This is because issues that would take you to a luthier's bench will cost more than the uke is worth in today's dollar. Cleaning, setting a head, polishing, even replacing tuners, you can do that at home for very little money, but a luthier doing any of these tasks will set you back.
Midrange ukes - Strombergs with resonators, all types of midrange and high-end Langes, Brunos, Dallas C&D, top line Slingerlands and J.R. Stewarts, Weymans, Lyon and Healy Washburn, Bacons and similar are all (with the exception of Bacon) priced $200-$750 these days. These are worth laying out money for fixing if you see issues like separation on the resonator, cracked or disconnected dowel, and fretboard wear, cracks or missing frets. Replacing missing tuners, head replacement, set-up - many of these jobs can be done by you and should be. They're fun to do!
Gibson, Ludwig, and Abbot are top of the line - prices $600 to 3K+, and in restorable shape are worth acquiring as the money you put into them, you will likely eventually get back in value. The only exception on pricing is the UB-2, which I've seen go for between $300 in good condition and $700 in excellent cond. UB-2 could be your best value among these top of the line instruments. Without the mother of pearl decoration or sunburst finish, they are less sought after by collectors, but sound exactly the same as a UB-3. I was looking for one of these when I found my Ludwig.
All vintage banjo ukes are sopranos. No one set out to make 'concert' or 'tenor' instruments and the standard fret board is 15-17 frets at this time, even on the smallest cheapie. Longer necks, 19+ frets, were made by Lyon and Healy, Lange, and Weyman most commonly. These were not called concert instruments, but rather were called 'longscale' ukes in the company catalogs. Creating different flavors of banjo uke is a relatively new development - I think Gold Tone was the first to do this, and they did it within the last decade. It was common in the 20s and 30s to tune tenor banjos in Chicago tuning - the same as a baritone uke - but that's about it.
Probably more than you wanted to hear from me, but I'm trying to look busy today.
