Oddly, not being able to stand up on the board is often not entirely or even mostly because you have bad balance, its because your body isnt used to what is going on and doesnt know how to cope. No muscle memory or understanding. Tightrope walkers will fall off surfboards just as much as us normal types. You can only cure that by surfing more, and eventually you will realise you are standing with no balance problems. That said, improving balance, strengthening relevant muscles etc has no downside to it and a lot of upside.
There are 'surfing aids' like indo boards
http://tinyurl.com/ne97wxt that do help balance, but personally I dont rate them. I'm sure they improve your balance, but there are other ways to do it much cheaper and indo board arent really very close to surfing anyway. In my opinion. That said, indo boards can be fun in themselves (and of course skateboarding is another balance exercise that has its own fun). You can make your own indo board if you want for much less than they cost to buy. Just dont practice in front of the TV screen.
But outside of those external aids, and understanding that more surfing will often 'cure' the problem:
First: think about the muscles that allow balance - you are mostly talking ankle (strength and flexibility) and glutes. Do exercises to strengthen the muscles, keep your ankle flexible (as a teen you probably are flexible anyway). Your core is always worth working on. that said, if you are a fit/active teen, you should be in pretty good shape
2nd, there are exercises that combine strength and balance, my favourite is the single leg romanian deadlift or RDL, plenty of google examples. Pistol squats/one legged squats or even lunges are others. These are good exercises for general life as well.
3rd, and probably most relevant, do balancing exercises like standing on one leg for a while, then lift your unsupported leg so the knee is above your hip. then stand on something a bit unstable (like a round log, no need for a bosu ball or anything too expensive). Then do it with your eyes closed. Stand on one leg and have something throw a ball at you to catch, or wave your arms around, twist your body etc. Wear a heavy backpack and stand on one leg. When you are in an elevator, stand on one leg just for fun. Continually put yourself in unstable situations. Do the one legged exercises mentioned above.
Finally, your surfing technique needs to be considered - for example, is your pop up smooth or jerky; do you pop up with bent legs and centred balance or are you straight legged or in the 'poo stance' (see this useful video
https://www.youtube.com/watch?v=6rsq2Px8_JQ_ ). Obviously away from the water this is a bit hard to work on other than dry land pop ups (which arent ideal), but will definitely be an in-water factor to think about. Popping up low, hips below shoulders, knees bent, weight evenly distributed - if you dont get this right, you will fall over.
If you have a big break between surfs you definitely get rusty and take a few (or more) surfs to get back into the groove - so understand it will be 2 steps forward, one step back most of the time. Nothing you can do about it except move house!