I say 1/2-3/4inch thick plywood , I sat a minimum of 4 sheets ( one for top,1 for back 1 for floor and then one of sides( depending how wide and tall you make it ) You can also possible look into sealing the cage with a resin that water proof once dried ( I say to sand it smooth and apply a second coat if this will become a life time cage to make sure it sealed good. I know when I built a 4x2x2 cage I sealed the wood with a roll on bed liner paint and it was great ( I even managed to have my heat tape under the liner/paint so it was protected from moisture.) It took 2 people to move it and then it was hard to move in my mom's trailer cause the hallways was narrow left little room for errors.
I do think if you do this make everything and paint/ resin coat it with plenty of air. and then assemble in the final resting place. so you only have to go over joints to make water tight. I don't know if I still have the pictures anymore to show my finished cage. I know when I rehomed a couple bps that one cage went with the snakes and I last heard still using them.