Tenor Banjo Bridge Height
 

I'm assisting my boss with getting parts for putting back into commission, a 1900s to 1920s tenor banjo he inherited from his father. It's missing a bridge, and I discovered that StewMac sells 4-string banjo bridges. The thing is, they offer them in 2 different heights:

9/16"
5/8"

Which would be the appropriate height bridge to get?

Also, would it be metal string, or is there a possibility, that it would need to use gut, or nylon strings?

Impossible to determine without knowing the neck angle (relative to the head plane) on the instrument. Many if not most 100 year-old banjos have an improper angle resulting from years of improper storage and maintenance.

Buy both, and if you're lucky one will work without having to shim the neck (not in itself a particularly difficult thing to accomplish), but unless you're experienced you won't know the prognosis without trying a bridge.

I’m not too experienced with tenor banjos I got my first very cheap one earlier this year. But take the deflection of the skin into account.

Anyway, do yourself a favor and get it into service. It’s SOOOO MUCH FUN!!!
I use mine in the band for a few songs and it’s like a time machine bringing you back some eighty years. Easy to learn too.

Photos of the tenor in question would help but it's most likely that it took steel strings back in the day. The string gauges you use would depend on whether or not you'll be tuning it CGDA or GDAE. Just a heads up that the Stew Mac bridges can be a bit hit or miss - I got a couple from them a few years ago that split under pressure when I started tuning up, so definitely order a few of them so you have back up if that occurs. As already advised here, get both heights and then you can swap them out to see which one works best.

It is probably a 99+% chance it is built for steel strings but nylon would not hurt it if he wanted to play quietly. It would have been built for CGDA tuning as the "Irish Tenor" technique started around 1960. If he is interested in playing jazz and strumming styles this is what he likely wants.

In the 1920's, most banjos were designed for 1/2" bridges, make sure you get a 4 string bridge. Over time banjos get set up differently and a taller ( or lower) bridge may now work better.

Bridges can be bought many places for about $5 and up, even Amazon and better music stores. Elderly music may be a good choice as Stew Mac is not the dedicated banjo supplier they were in the 1980s. Since they are cheap, I suggest you buy 2 or 3 sizes and see which works best. You can sand the bottom to slightly lower if needed.
Pics of the banjo would be helpful.

My Kent tenor banjo used the 9/16 bridge. You could start there, 50/50 chance.

I installed a 5/8 Bart Veerman "keltic" bridge on my tenor which gives it a good sound. I ordered it from Elderly which has tenor bridges for as low as $8, and in various heights. I tried some higher bridges that I already head and I didn't like the sound. The Veerman was $30 but in my opinion was worth it.