Form follows function, or that's how the saying goes. Which isn't true for some things. That is to say, a large reason why humans use a base 10 numbering system is because we have 10 fingers, not that we have 10 fingers because we use a base 10 numbering system. Except as worldbuilders, we get to do the reverse and create an alien race with the precise number of fingers that would be the most useful.
In my sci-fi setting, I'd like the most advanced race to have an extra advantage - a superior numbering system, using base 16 instead of base 10. 16 is a perfect square, almost a perfect number, and uses squares of 2 as it's landmarks of higher numbers (it's 'ten's place' is 256, it's 'hundred's place' is 4096, etc.). It handles larger numbers far better than a Base 4 system would, and easily converts to binary.
So now comes the real question - how do I design the aliens such that a base 16 system is organically produced? After all, number systems aren't designed in the modern. Eight fingers on one hand is too much for my tastes, and a four-armed, four-fingered alien on two feet doesn't seem like it would naturally arise. And, considering this the 'smart' race, I'd rather not have a solution that comes up as a result of combat with each other, something which the four-armed variation definitely suggests.
Two points - one: Yes, I know there are ancient 16-bit numbering systems that have survived to the modern era. However, they don't see widespread use today. I'm looking for a way so that I don't have to handwave and say they just developed a system like that and took over. Second - an answer which proposes a different base system will be accepted, so long as you can definitively prove it superior to my base 16 for the purposes of a sci-fi setting.
EDIT: Reminder, I'm not asking which Base numbering system is superior. As far as I'm concerned, they've all got strengths and weakness, and 16 is what I find easiest for my setting. The question is a xenobiology question about alien design and development.
EDIT II: The most common numeral system used currently, and historically, is the Hindu-Arabic numeral system, as it is so called. It is Base 10, and spread throughout the world to become the backbone of math, which has only recently begun to commonly use other bases because of programming.