Ilya Somin at The Volokh Conspiracy discusses giving children a vote.
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.
I have some objection to giving an intelligence test because if applied to adults it is unconstitutional, but I am willing to entertain arguments that there is something special about age to change that determination.
Ilya Somin also points out:
Some commenters note that children might lack maturity or life experience, as well as knowledge. Obviously they do lack it. I'm just not convinced that either is tremendously useful for voting. Most voting decisions have to do with complex, large-scale policy issues that can't easily be weighed based on personal experience. Realistically, even most adults have little life experience that is directly useful in assessing difficult policy issues.