Sure, it's currently one of the best written governing documents that humans have historically produced, but nothing is perfect and you have to be really careful about it. How about just finishing what Roosevelt proposed back in 1944? A second Bill of Rights. It still seems to be a pretty good idea.
The only problem I see is the major push for a "constitutional convention" to actually change it is being driven behind the scenes by monied interests that would love to figure out how to charge you for breathing. I love libertarian ideas of personal responsibility and accountability for your actions, as well as the NAP; but the behind the scenes stuff that is funding some of the rhetoric on the national stage is pretty subversive, and the "Tea Party" is a perfect example. Monied interests would love for you to vote in more 'freedom' so they can charge you more, or start charging you for something everyone currently gets for free.
So I would be really careful on how and what is allowed to happen. Specifically because of history, as mobs are not thinking individuals discussing things; they just gather around something so they can throw in anything they disagree with, and watch it burn. Not even comprehending if they have been manipulated to burn their own protections.
Some of the christian religion factions (Dominionism especially) have been trying to subvert the first amendment as soon as it was signed. No better than any of religious theocracies around the world. Which is why the first amendment was put into place in the first place. And over the years the religious folks have been successful in changing America's motto, pledge, or adding their creeds to the money, etc; mainly due to their erroneous 'persecution complex'. But they are all surreptitious violations of "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances." Which is there to protect their religious views in the first place.
To have this discussion, you really need to really discuss it.