I like SamNavy's approach to arguing this point. It think this approach can win fence-sitters.
I also point people to the following argument, which was floating around in the months leading up to the Heller decision.
Suppose the Constitution provided:
A well educated Electorate, being necessary to self-governance in a free State, the right of the people to keep and read Books, shall not be infringed.
This provision, which is grammatically identical to the Second Amendment, obviously means the following: because a well educated electorate is necessary to the health of a free state, the right of the people to keep and read books shall not be infringed. The sentence does not say, imply, or even suggest that only registered voters have a right to books. Nor does the sentence say, imply, or even suggest that the right to books may be exercised only by state employees. Nor does the lack of identity between the electorate and the people create some kind of grammatical or linguistic tension within the sentence. It is perfectly reasonable for a constitution to give everyone a right to books as a means of fostering a well educated electorate. The goal might or might not be reached, and it could have been pursued by numerous other means. The creation of a general individual right, moreover, would certainly have other effects besides its impact on the electorate's educational level. And lots of legitimate questions could be raised about the scope of the right to books. But none of this offers the slightest reason to be mystified by the basic meaning of the sentence.
The Second Amendment is no different. Modern readers may have difficulty in seeing how a general right of individuals to keep and bear arms could contribute to a well regulated militia and to the security of a free state, and we shall explore that question in more detail below. But the text of the Second Amendment offers not the slightest warrant for presupposing that the answer to the question is that its framers were semi-literate fools who meant to say something like "The states shall have the right to maintain independent military forces for use against the federal government."
I have had good luck with this line of reasoning as well... I think this line of reasoning started with Robert Levy of Georgetown University.