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... 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.