Technically, you should be "chasing" the threads, not re-tapping.
Chasing threads is the act of using a tap to repair damaged threads and remove dirt, etc., from them, not create new ones. If you try to re-tap a threaded hole using the wrong tap say, a 3/8"-24 tap in an M10-1 hole, you'll be left with a very weak threaded hole that you'll probably strip the first time you try to tighten a bolt in it.
If you don't know what the thread is (and in this case, you don't), use a T-handle to hold the tap and turn it using very light hand pressure only. If you feel serious resistance, stop--that means you're cutting metal and probably don't have the right tap.
Use what is called a 'bottoming' tap as they will chase more threads in a blind hole and offer more resistance when the tap's threads don't match the hole's threads.
If you can't repair the threads this way, you can use a threaded insert to repair the threads. Auto parts stores sell these; if you ask for a Heli-coil repair kit they'll know what you need.
Just a FWIW, probably not applicable in this case, there are other thread types. Whitworth is one but you usually don't see this in American equipment. Only place I've encountered Whitworth is in old English motorcycles.