To put it another way, we let the mentally retarded and the mentally ill vote, and those votes are not weighed more heavily than those of (for instance) people holding doctorate degrees. We would be hard-pressed to justify this policy on the basis of a more knowledgeable electorate or better outcomes. If we wanted to increase the average knowledge of the electorate, there are many changes that we could plausibly take in addition to letting children vote -- literacy/political/economic/history tests for example.* The fact that we don't have these tests, and that we find the idea generally repugnant and elitist (in a bad way), seems to indicate that we value democracy not for its ability to generate the optimal outcomes, but in that it gives every person an equal say in political decisions. The idea that democracy does not necessarily lead to optimal outcomes is very old -- in The Republic, the democratic man as drunken, incontinent, and like a sleepwalker.
Basically, there may be good reasons to allow children to vote, but a justification based on improving the knowledge of the electorate seems to me fundamentally incompatible with democracy. The real question is: are children affected by the decisions of government? If so, why shouldn't they get a say in the matter?
*Let's set aside for a minute the historical racist use of voting tests.