Start with premise: 48% of US taxpayers presently have a Federal tax liability of $0. But polling on replacing the income tax—like national sales tax, VAT, etc show pretty strong support. I know, crazy, right?

If I wanted to make it popular, I'd do it this way:
Set a fairly high value added tax rate. Maybe not as high as Denmark's 25%, but 20% seems to work most places.
Take maybe a quarter of the revenue, and send it out to everybody on a per capita basis, on a super easy opt-in basis. (opt in because I suspect a lot of thoughtful richer-than-median people could easily be dissuaded from asking for theirs). So, round numbers, the amount you pay on this tax is a percentage of what you buy minus a constant. If you don't buy very much, as with a substantial fraction of the population, you get a net gain. For those who think it might be too much, some of it could add funding to earned income tax credits, one of the very bright spots in US tax policy.

The reason isn't just to make it popular, of course--it's to make sure that it doesn't hit lower income households disproportionately. Adding a brand new tax burden to those folks wouldn't be very nice.

Also, having a big "refund" tier is the best bulwark against the inevitable endless lobbying for exemptions. The tax should, as far as practicable, be applied on *everything*, including necessities. (there is a lobby for everything, and everything is therefore a necessity). You get massive tax fiddling and wasted economic effort at every artificial boundary you introduce. Is this food or a snack?

As an aside, the best way to hand out the money would be debit cards tied to "everybody gets one" accounts at the central bank, recharged monthly from afar. This has the advantage that in times of crisis, helicopter money could be sent out on a day's notice.

Jim