The Aztecs needed to have wars from time to time, for religious reasons (they needed the captives for human sacrifice) and social ones (it gave ambitious young men a chance to prove their mettle). When they had no reason for a real war, on occasion, they would arrange a 'Flower War' - two cities would
agree to hold a battle over nothing whatsoever, purely for the useful side effects.
The Space Race had a lot of good side effects. It made people care about science, it gave the government an excuse to spend more on education, and it developed oodles and oodles of spinoffs - the entire Internet can be considered a spinoff, and certainly materials science advanced by leaps and bounds. Okay, the direct results weren't useful, but the Space Race would still have been a good thing for the USA even if they had never landed on the moon. Yet in modern times, space exploration is badly neglected, regarded as a 'prestige' project with only psychological value, that needs to be shoved aside. And it can be shoved aside, because we're not trying to beat the damn Commies anymore, right? What's the point of space exploration once it's become a matter for international cooperation?
Clearly, the USA needs someone to race with again.
I'm thinking India. Why India? Well, they have a serious space program already; they've sent a lander to the moon, and are contemplating a Mars launch by 2015. They're a growing economy with a huge talent pool, and could probably have very impressive space program if they wanted to. An effective space race could provide benefits to India via spinoffs and increased educational investment, much as it did for the USA. China might also work - they've already launched an astronaut - but the political aspect could be a lot thornier. Brazil might work, as well. Russia is probably out, since the USA is relying on Russian launches to get at the International Space Station in the wake of the Space Shuttle's retirement, but they might go for it anyway for old time's sake.
But whoever the USA makes arrangements with, the governements will make a friendly cooperative announcement that they are holding a Space Race. For the advancement of science and the chance for vast prestige and glory, they will compete for some distant goal, like landing a robot on the moons of Jupiter, or for no particular goal but with an attempt to simply go wherever their research takes them within the next, say, fifteen years. Since the USA already has a developed program, perhaps there's a handicap - the other player gets free access to NASA's files, or some major cash grants, or the right to launch from Cape Canaveral for free. Any joint programs will continue. In any case, the announcement is made in a sporting spirit.
The countries then proceed to develop their space programs with dead serious warlike fervour.
The cause of Science is advanced. Economies are stimulated. Spinoffs are developed. Patriotic spirit is increased. People get to follow something a lot more exciting than the World Cup, complete with explosions.
Winner gets the right to go 'neener neeener' at the other.