The big advantage would be that you learn to see that there is nothing magical or 'right' of one number base over another. Base 2 has it's uses as does base 10(10). And most computer programmers are familiar with at least one other base besides decimal, and quite a few will be able to use binary, octal and decimal with relative ease. It gets interesting when you go off the beaten path and you re-learn the rules for arithmetic in different bases. In my experience all of this gives you a much better understanding of why decimal is practical and widespread. But it also shows you that it is a convention that won out for both cultural reasons and because it made certain arithmetic easier.

If 10x10 = 100 looks natural to you in decimal then it will still be natural to you if you think of it as 16x16 = 256 when you look at the numbers in their hexadecimal representation. You can only get that kind of fluidity by playing around in different number systems. So I'm perfectly ok with students inventing their own number systems, they are definitely not going to get any dumber on account of having done that.

Growing up with Inches and degrees Fahrenheit is a cultural issue, most of the rest of the world has moved on from there, for reasons that are far more compelling than those that would apply to using a different number base. Those are arbitrary values, whereas all number bases exist regardless of whether we use them or not. Think of the one as cultural baggage and the others as just another part of number theory.