what drill bit size to pilot drill hole into stud for 15/64" lag bolt
When you're drilling and tapping threads into metal to accept a machine screw, you have to use the precise right size drills and taps.  But you don't need to be real precise with lag screws in framing lumber.  Without a pilot hole or with a too-small pilot hole the wood may split.  If the pilot hole is too large, obviously the fastener won't hold.  

Look at the lag screw, observe the diameter inside the threads  (R, the "root diameter") in the illustration below) not the thread diameter (T).  Match up a bit that's slightly smaller than the root.  It sounds like you have 1/4" hardware and a 3/16" bit would be fine.