I would modify Fabio Rojas's answer ever so slightly by saying that the subsidy should probably go to child-rearing rather than child-bearing. My impression is that the marginal non-parent is worried not about the hospital costs but the costs of child care. Child care subsidies would probably be more effective than lower out-of-pocket hospital expenses, but I could be wrong (if you're a grad student thinking about what you're going to study, notice that there's a paper here).
So here's my answer: subsidize the development of quantum and biomolecular computing, perhaps with large prizes (as Daniel Kuehn suggests). Why? Computing is a general-purpose technology that would make presumably just about everything we do much cheaper, including health care, energy development, retail, and all sorts of other things. It's currently being developed by commercial enterprises, but even a speed-up of just a couple of years--or even a few months--might be more than worth it for the paltry sum of $1 billion. This would be about equal to 1/7 of the NSF's 2013 appropriation.
Kaldor-Hicks: it's close to a general-purpose "public good" technology. Readers of Kahneman and others know that our information-processing capacity is not so great. More information-processing power means better decisions.
Utilitarian: Bryan points out that we should consider the preferences of people who will appreciate the endeavor, perhaps in an aesthetic sense. Pure science offers a lot of nonpecuniary "isn't this awesome?!" benefits. Bryan also mentions a preferential option for the poor given the diminishing marginal utility of wealth. Here's where much more rapid, much cheaper information-processing technology really benefits the poor. Here's why. People are poor in part because of very bad decisions. At the margin, better information-processing technology should benefit the poor more than it benefits the rich as it might help them overcome and better understand the costs of impulsiveness and lack of conscientiousness. I admit this remains speculative, but better-developed quantum computing would, I think, bring us closer to the day when we have Apps for Bayesian applications, continuous-time data analysis, and continuous-time updating of a dizzying array of probabilities.
Here's one example. Sites like Match.com and eHarmony.com will develop better matching algorithms, leading to better matching on preferences and characteristics that matter and, presumably, less wasteful signaling in the marriage market and lower divorce rates. Less divorce means less poverty, less emotional devastation, and fewer social problems if divorce has negative effects on the next generation.
If you've gotten this far, you probably have all sorts of criticisms. So here's one question for readers: why won't this work?
And here's a bonus question for readers, based on a conversation I had with Bryan about a year ago: In the short run, Facebook will probably lead to an increase in the divorce rate. In the long run, it will probably lead to a reduction in the divorce rate. Why?