Rationally, my vote makes little to no sense. There's virtually zero chance that my vote will make any difference, or even that my circle of influence could be large or concentrated enough to swing the outcome of even a county bond issue anywhere. So why did I? Here are two possible secular answers, and then my somewhat more involved theological answer.
First, there's the classic warm fuzzies that you get from being part of something much greater than yourself. It's a primal human urge to align yourself with a movement or cause that is transcendent and powerful, and voting can help satisfy that longing. I don't think there's anything wrong with jumping onto that kind of psychological bandwagon; in fact, I think it's kind of silly to try to deny that you feel it. (If voting isn't the way you want to meet that need, that's fine, but don't pretend like you don't have the same desire that you express in one way or another.)
Second, I also think voting is just a part of being a good citizen. It also won't make any difference if I don't stop at a stop sign on a deserted street, and yet I should. Doing one's civic duty creates order and has larger-scale effects than the sum of the individual unimportant acts. I want to be a good citizen, so I vote.
Finally, and what I think is actually the most important reason for me personally, is the religious angle. Now, this is my personal interpretation, but I read Mosiah 29 to be telling me I ought to vote. The chapter talks about a political crisis where the people are begging for the current (righteous) king to appoint a successor, but the righteous king realizes that there's no good candidate available. So he explains to the people that wicked kings lead their nations to do all kinds of evil stuff (see v. 17 and 21-24). That's obviously bad, but the most interesting thing is that in verse 30, he tells them they shouldn't have a king so that "if these people commit sins and iniquities they shall be answered upon their own heads." In other words, he's telling the people to be responsible for their own screwups. To a lazy, unrighteous person like me, that sounds like a terrible idea--why not let the king take the blame for the nation's collective sins like verse 31 says would happen (if the king causes the sin, at least)? Free sinning, woohoo!
And yet, verses 37-38 tell us the people were quickly convinced by this logic, and that "every man expressed a willingness to answer for his own sins." Well, maybe these people are just better human beings than I am and are adults, but there's even more! After actually gathering together and voting, "they were exceedingly rejoiced because of the liberty which had been granted unto them" as verse 39 tells us. Why? They were opening themselves up to potentially more condemnation!
The chapter doesn't spell out the answer, so this is definitely my own doctrine, but it seems to me that concurrent with that potential for extra condemnation if they chose evil over good, they also opened themselves up to more blessings if they made their voices heard for righteousness and goodness--there are costs, but there are also benefits! This is analogous to the choice to come to Earth and exercise agency here. We can only gain much if we risk much. The people from Mosiah 29 were ecstatic about the opportunity to vote because they were free, they were agents unto themselves, and they could finally freely choose good. Being an adult sucks in that you have bills to pay and stuff, but it's also awesome in that you can drive where you want, go to the movies you want, talk to the people you want, stay out as long as you want, etc. The people in this chapter recognized all that and rejoiced. I read the Book of Mormon here to be telling me to do the same. If I want any blessings for trying to make the world a better place (as selfish as that sounds--there's no time to explain why I don't think it (necessarily) is), one way to do that is to vote towards that change. If I want to be a spiritual adult, I need to take responsibility (and also get credit) for my choices regarding my community. Naturally, voting alone won't avail me much at the last day, but I believe it will be one part of my judgment.
To be clear, I reject the idea that we need "to vote as [Jesus] would vote" if that means that there's one "correct" candidate and voting for the other guy (or gal) is a sin. Unless Hitler is up for county commissioner on your ballot, that's way too simplistic a view of the complicated/boring policy choices that candidates and issues represent. But I agree with that idea if you interpret it to mean you ought to vote according to your conscience and using thoughtful, prayerful preparation. In other words, I believe that voting as Jesus would vote would also have to entail not judging those who vote differently since he commanded us to "judge not." Vote humbly, meaningfully, and with love--that's how you vote as Jesus would vote, in my opinion.
So those are the reasons I think I voted. You may not think they're entirely rational (or maybe you do). But regardless, rationality is overrated, anyways. Who wants to go to a football game with someone who's going to act rationally? Because the rational response is to be utterly bored with 22 men running around in spandex for no reason. Just get over yourself and jump in and cheer and have some fun! That's my vote, at least.