We recently "enjoyed" a game that was played during a monsoon. The kids were losing their grip on their stick... and even their lax stick butts were coming off. :)
You should tape your sticks when things are dry. Tape the areas where your son needs the most grip (i.e. near the butt of the stick and/or the head). However, the tape position should depend on your son's likes and dislikes. We like to leave the middle of the stick tape free so my boys' hands are easily "slide" to another position on the lacrosse stick.
Try sports tape or spend a little more and get specialized lacrosse tape. FYI - We are cheap so we stick to the regular sports tape!
If you get the position wrong, it is easy to cut off with an Exacto blade, etc.
So before the next rainstorm, prepare by taping your son's stick!!