More often than not, when the top came off, the fitness of the diam of the bat was damaged. I mean damaged in the sense that the bat’s roundness is no longer round. It’s probably egg shaped … with somewhat of flat spot on it… along the length of the barrel, from the length of the 18" ring** to the part where the cap came off.
If you’re talking a very young baseball league, an inexpensive bat from a dept store will be a good replacement. I would not recommend repairing it with supper glue or anything like it. Also, check with the league that your son is in … they might have something to say about it.
One of my players had a relative who was playing 10 - 12 baseball, and the kid’s team went all the way to the playoffs, when it was found that better than half of their bats were illegal. From what I gathered it caused a big problem for all concerned. So, doctoring your son’s bat might be considered out of bounds … rule wise. Check with your son’s league.
** From the heel of the bat … its nob end, just below the hand grip, then measuring 18" up the barrel of the bat … is where pine tar or other gripping agents can be applied so the batter has an enhanced grip. BEYOND that 18" measurement … usually called the “ring”, the bat can have not substance(s) on it that might influence the bat’s performance when striking the ball.