First, get some Teflon (non-adhesive) tape such as used for plumbing, and by trial-and-error find just the right amount to wrap around the threaded-segment of the arm to snug it right up. Even if you replace the arm, it will probably be loose or soon become loose, and still need the teflon tape. Works great, used that for DECADES, occasionally needs to be re-applied but tha's alright, mama. It doesn't take very much tape. (Haahh!
I only just now saw Larry's suggestion above... !
If you really still need to figure out exactly and precisely what threads it used in the first place and replace it, there are thread-gauges that measure the threads by matching up to them; any good machine shop will have a number of 'em on hand. Maybe, hopefully, you've got a friend or a friend with a friend that is a machinist...