Ilya Somin, a law professor who was probably a bright kid, makes an argument for kids who can pass, for example, a civics test to be allowed to vote.
The main objection to giving children the vote is that they lack the knowledge to make informed choices. Of course the same is true of most of the adult electorate, who are rationally ignorant about politics and public policy, and often don't know even very basic facts. Nonetheless, it's probably true that the average child knows a lot less about politics than the average adult, and that may be a good reason to deny most children the franchise. But why deny it to all of them? If a minor can pass a test of basic political knowledge (say, the political knowledge equivalent of the citizenship test administered to immigrants seeking naturalization), why shouldn't he or she have the right to vote? Such a precocious child-voter would probably be more knowledgeable than the majority of the adult population. Giving her the right to vote would actually increase the average knowledge level of the electorate and thereby slightly improve the quality of political decision-making. I've met twelve-year-olds with far higher levels of political knowledge than that of the average adult. You probably have too.
Once the knowledge objection is off the table, all the arguments for giving adults the right to vote also apply to sufficiently knowledgeable children. Like the adults, children have a claim to the franchise because government policies affect them too, because otherwise their interests might be undervalued in the political process, because it affirms their status as citizens with equal rights, and so on.
Obviously, there might be some difficult administrative issues. For example, we might not trust the government to put together an adequate knowledge test. But I don't see any principled reason to deny the franchise to children whose political knowledge is greater than that of most adult voters. [The Volokh Conspiracy - Should (Some) Children Have the Right to Vote?]
Except for the fact that it's in the Constitution, which would have to be amended, I don't see a principled objection to it. And the Constitution has been amended in my own lifetime to change the voting age.
In Robert Heinlein's Starship Troopers, which famously suggested limiting the franchise to people who were willing to perform a term of military or service, Heinlein has a "History And Moral Philosophy Teacher" say
All systems seek to achieve [stable and benevolent government.]by limiting franchise to those who are believed to have the wisdom to use it justly. I repeat 'all systems'; even the so-called 'unlimited democracies' excluded from franchise not less than one quarter of their populations by age, birth, poll tax, criminal record, or other.
Major Reid smiled cynically. I have never been able to see how a thirty-year old moron can vote more wisely than a fifteen-year-old genius . . . but that was the age of the 'divine right of the common man.'
But Ilya Somin hasn't mentioned the main obstacle to this system—"disparate impact" theory. If we started giving civics tests to fifteen year olds I can guarantee you that whites and Asians would pass them more often than blacks and Hispanics. And that means that Supreme Court would come down heavily on it, as they always do when there's high stakes test that displays racial disparities.