This idea was impossible just a few decades ago. It simply wouldn't have been feasible for all taxpayers to have a say due to technology constraints. But the computing power available today would actually make this pretty easy. It could be like choosing your 401k fund distributions. Each person could just provide percentages that add up to 100% of their taxes for various categories provided by the government. This could be done quite easily online or on paper as part of your tax return.
As a demonstration of how easy this would be, here's how it would work with the list provided by Third Way (which is admittedly incomplete, but a start). I added the final row for debt paydown.
You can see that the government can provide a "Suggested Contribution" section, but it then allows people to choose which areas of spending they like and don't like. In this case of the example above, whoever filled out the card dislikes government-sponsored health care research, education funding, arts funding, and Amtrak. But she is very concerned about the national debt, so doesn't simply want it to roll over. A different taxpayer might have had opposite priorities.
I know what you're thinking: there's no way. Chaos would ensue. How would agencies know how to budget for the coming year? That's easily solvable. One solution could be creating a lag of a year or two between when tax allotment is made and when it is actually used. That way, agencies will be able to plan their budgets accordingly.
So what happens if, say, Social Security becomes underfunded? It reworks its budget. It cuts benefits or revises its strategy, say to eliminate distributions for retirees above a certain net worth ceiling.
The biggest objection is that important programs will be underfunded. But that consequence is actually logically impossible. If Americans don't think a program is important enough to provide more funding, then in a democracy, by definition it isn't a priority so doesn't deserve more funding. I also think you would be surprised at how much money Americans would choose to provide to entitlements. More Americans might want contribute money to Social Security and less to the wars in Iraq and Afghanistan, for example, because they might prefer to provide for the retirement of Americans instead of bombs in the Middle East.
As you can see from my sample you could also include checkboxes for options like using tax proceeds wherever it is most needed. This is a common option provided for donations to nonprofits, but in this case you would be essentially giving Congress discretion to spend your money however it chooses. Of course, that's what we all do now.
A few changes would be necessary to make this work. First, at least one category would be non-discretionary: interest on the national debt. There's no getting around that one. Second, you would need a balanced budget, so eventually that would be phased out anyway. If the government can borrow, then it can juts continue to pay for programs that Americans don't support and increase the interest payments it forces on taxpayers. Americans could also choose to donate to an emergency fund, for things like wars, financial crisis, etc. That way, the government wouldn't need to be reliant on borrowing in such situations.
It would be fascinating to see how a discretionary tax payment system like this would work. Remember, the total amount of taxes people pay wouldn't change -- they'd just be able to demand how their money is spent. And anyone who doesn't want to be bothered distributing their taxes themselves could simply check the "suggested contribution" or "wherever most needed" box. Activists, however, could choose to give more or less money to whichever government programs they find more or less important. Of course, a less aggressive program could also be employed instead to provide taxpayers discretion over only a portion of their taxes and certain programs.
Sadly, there's approximately zero chance Congress would be willing to give up its discretion over how it spends taxpayer money. After all, that's its favorite pastime. So unfortunately, this will never happen. But it certainly would make Americans feel better about paying their taxes.