I find that motivating algebra is best done with cyclic groups for the following reason.

Students under a certain age - whether or not they care much for math - seem to like just about anything that involves breaking down their old ideas about math. They like seeing that what they took for granted or was simply taught as the truth is really just one [arbitrary] choice.

Introducing finite cyclic groups as simply a new way of adding the numbers they've been dealing with for years seems to be pretty stimulating. For one, they can add in small cyclic groups simply by drawing arrows around a circle of n points. If you're decent at drawing, there's also the intuitive bonus of describing this new way of adding numbers as moving around a helix that was once the regular integers - here I mean thinking of the integers as an infinite bit of string and wrapping it around itself.

I've found this sort of thing is very interesting, even for the most disaffected students. From there it's pretty easy to motivate just about any other reasonably tangible algebraic structure as "just another way of adding."

Anyway, my point is that the idea of algebra (and analysis I suppose, but I try not to think about it) as something anarchic is a powerful tool when getting through to college students. Confirming their middle/high school suspicions about "all of this crap being arbitrary" is psychologically very comforting, and from there you can go on to tell them that you get a whole host of interesting ideas simply by changing your point of view.