Are we allowed to use anything besides the balance? Do the balls all look the same, are the same size, ect?
If so, this is a tough one. With three tries, binary selection isn't going to work (2^3 is only 8). My first thought was to only 6 on the scales, if the scales change, you know its in those 6, if it doesn't you know it's in the other 6. Then put 4 on, if it stays the same you know its one of the other two, if they change you know it's one of the pairs on there. If it was the pair not on the scales, then you can put a "normal" ball on the scales with one of the two that you've narrowed it down to, and that will tell you which was the ball you're looking for, but as you can see this won't work.
My second thought is to break it into 8 and 4, which might have a shot. (kinda thinking as I type here) If we put 8 balls on and they balance, we know its one of the remaining 4. Then we could just swap 3 of the remaining 4 onto one of the sides, if they balance, we know the last one is the odd ball and can balance it against a normal ball, if they don't balance, we know we either have a heavier ball or a lighter ball and the 3 that it is in. We can then take 2 of the three and balance them, if they're unequal, we have our ball, and if they're equal, the 3rd is our ball.
If the 8 balls dont balance..... hmmm
One side either contains a heavier ball or a lighter ball, so... we ....(pause for thinking lol)
Swap out 3 of the balls on one of the sides out with 3 of the non-odd-balls, and swap the remaining one on that side with one on the other side. If the scales balance, then you know that your ball is in the 3 you removed, and you can clear off all the balls, pick 2 of your 3 that you know its in. If that balances, you know yours was the 3rd and from the inital balance you know it is either heavier or lighter depending on which side it was on. If it doesn't balance, well, you still know you're looking for either the heavier or the lighter one from the initial balance, and the heavier or lighter one in this balance will be your ball.
If the scales after the swap didn't balance, they either stayed the same, or switched balances. If they stayed the same, then the ball was one of the 3 you put on. Based on which side (heavier/lighter) you swapped the balls onto, you can tell that your ball in the 3 you swapped on was heavier/lighter corresponding to that, so pick 2 of the 3 and if they balance, the last one is your ball and is heavier/lighter depending on the previous balance, and if they don't balance you know which one is heavier/lighter already so can just pick that one of the two you balanced. If the scales switched balances after the swap, you know your ball was one of the two you switched sides on during the swap, since there is only 1 heavier/lighter ball and if it was on the opposite side to begin with, the only way for the scales to switch up/down would be for you to switch it to the other side. You then know whether the ball you're looking for is heavier/lighter, and can just balance out the two you had switched sides to find the one you wanted.
I think that's it! Better Check though
8/balance/
None.