I assume there's few (any?) people here in favor of Bush's Social-Security plan, so I'll skip over the usual reasons to oppose it since we all know those. Instead, consider what you might think if you considered yourself "fiscally conservative", maybe even a moderate Republican. Would you support it then? I'd argue that even if we grant a number of assumptions underlying that worldview, it's still a bad plan.
(Said argument inside.)
So, let's assume, for the sake of argument, that social security needs to be reformed, that taxes are already too high, and etc.
Now, is a good solution to take a certain percentage of the social-security-tax money and put it in private investment accounts? Well, no.
Taking one possible scenario, say we allow people to do whatever they want with the money in that account, investing it in any legal way (i.e. they can't "invest" it in a yacht, but anything else is okay). This comes with a reasonably good chance that some number of people will end up losing much of the money in their accounts. Not only can the stock market tank (as many people have pointed out), but there are a lot of very risky investment options available to people who are so inclined: gambling on penny stocks, options trading, currency speculation, and so on.
So what exactly is the benefit to this at all? If we want to allow people to invest for their retirement however they choose, we could just cut social-security taxes, which would be a lot simpler and have less overhead, and let them put the money into an IRA themselves (perhaps raising the IRA contributions limit). Forcing them to put it in an investment account perhaps will increase the overall savings rate, but there will still be a lot of people who squander that savings through either stupid investments (the investing equivalent of buying a zillion lottery tickets with it) or simply poorly-chosen ones.
If we're going to come up with a list of "safe" investments they're allowed to choose, then it seems to defeat the entire purpose of giving people control over their accounts. "You can invest in anything you want... so long as it's not [a, b, c, d, ...]". Not to mention the extreme difficulty in actually determining what investments are reasonably safe. If we're going to basically only let people invest in some sort of generic mixture of mutual funds and bonds, might as well just let the government handle it.
So, in short, no matter what you think about saving for retirement, this seems like a bad plan. If you think people ought to control saving for their own retirements, support cutting the payroll taxes to let people have more money left over to save, and increase the IRA contribution limits. If you think there ought to be a guaranteed pension from the government, then this plan obviously doesn't help with that. Taking some of the payroll taxes and funnelling them to special investment accounts is worse than both the main alternative options: Either give me the 1% back and let me do with it as I see fit, or keep it in the system and fund an insurance-like program; don't "give it back" to me with a million strings attached in some special account I'll probably have to file pages of paperwork about.