An argument that is sometimes made for a monetary policy that targets a path for nominal GDP (NGDP) is that it reduces risk for most borrowers who take out debt contracts with repayments fixed in nominal terms (see, for example, Nick Rowe here). However, as far as I am aware, this argument has not been quantified in a way that allows it to be compared with the more familiar benefits of inflation targeting. A recent paper by Kevin Sheedy does just that.
Before getting to the punchline, it is worth setting out the argument more precisely. A good deal of the borrowing that goes on in the economy is to smooth consumption over the life cycle. We borrow when young and incomes are low, and pay back that borrowing in middle age when incomes are high. To do this, we almost certainly have to borrow using a contract that specifies a fixed nominal repayment. The problem with this is that our future nominal incomes are uncertain - partly for individual reasons, but also because we have little idea how the economy as a whole is going to perform in the future. If the real economy grows strongly, and our real incomes grow with it, repaying the debt will be much easier than if the economy grows slowly.
As most individuals are risk averse, this is a problem. In an ‘ideal’ world this could be overcome by issuing what economists call state contingent contracts, which would be a bit like a personal version of equities issued by firms. If economic growth is weak, I have a contract that allows me to reduce the payments on my debt. However most people cannot take out debt contracts of that kind, or insure against the aggregate risk involved in nominal debt contracts. We have what economists call an incomplete market, which imposes costs.
Monetary policy can reduce these costs by trying to stabilise the path of nominal GDP, because it reduces the risks faced by borrowers. Of course monetary policy cannot remove the uncertainty about real GDP growth, but if periods of weak growth are accompanied by periods of moderately higher inflation, then this is not a problem from the borrower’s point of view. (Koenig discusses this in detail in a paper here.)
How do we quantify this benefit of NGDP targeting, and compare it to the benefits of inflation targets? Sheedy defines a ‘natural’ debt to GDP ratio, which is the private debt to GDP ratio that would prevail if financial markets were complete. Under certain conditions the natural debt to GDP ratio is likely to be constant, and Sheedy suggests that departures from this benchmark are unlikely to be great. So a goal for monetary policy could be to close as far as possible the gap between the actual and natural debt to GDP ratio, in an analogous way to policy trying to close the output gap. To do this it would target the path of NGDP.
The current standard way of modelling the welfare costs of inflation, due to Woodford, is to measure the cost of the distortion in relative prices caused by prices changing at different times to keep up with aggregate inflation. This suggests monetary policy should have an inflation target rather than a NGDP target. (I note here that typically in the literature these costs are far greater than costs associated with output gaps.) What Sheedy does is set up a model which has these costs of inflation present, but also has the costs of nominal debt contracts discussed earlier. With these two different goals, an optimal monetary policy will go for some combination of inflation targeting and NGDP targeting. The key question is which kind of costs are more important. 
Sheedy’s answer is that the costs of nominal debt contracts are more important. The optimal monetary policy gives a 95% weight to the NGDP target, and just 5% to the inflation target. Now of course this is just one result from a highly stylised model, and Sheedy shows that it is sensitive to assumptions about the duration of debt contracts and the degree of risk aversion. Nevertheless it is very interesting result.
A very simplistic way of describing why this may be very important is as follows. If the focus of monetary policy is always on the cost of inflation, NGDP targets will appear to non-economists at least (e.g. politicians) to be second best. They are a nominal anchor, so we will not get runaway inflation or deflation by adopting them, but why not just target inflation directly? Who cares about nominal GDP anyway? This paper suggests a simple answer - borrowers care. If we see monetary policy has being important to the proper functioning of financial markets, as we now do, then reducing the risk faced by borrowers is a legitimate goal for policy. It may make sense for inflation to be high when real growth is low, and vice versa, because this reduces the risks faced by borrowers. I think a politician that was not beholden to creditors could sell that.
 For those interested in government debt, there is an interesting parallel in the literature. Some authors (e.g. Chari, V. V. and Kehoe, P. J. (1999), “Optimal fiscal and monetary policy", in J. B. Taylor and M. Woodford (eds.), Handbook of Monetary Economics, vol. 1C, Elsevier, chap. 26, pp.1671-1745.) developed the idea that nominal government debt contracts could be a useful way of avoiding costly changes in distortionary taxes following fiscal shocks, because inflation could change real debt. However Schmitt-Grohe and Uribe (Schmitt-Grohe, S. and Uribe, M. (2004), “Optimal fiscal and monetary policy under sticky prices", Journal of Economic Theory, 114(2):198-230) showed that once you added in nominal rigidity to the model so that inflation was costly, inflation costs dwarfed any gains. This example makes the fact that we get the opposite result with private debt contracts particularly interesting, although as Sheedy and others have noted, this may be partly because this earlier literature assumed short maturity government debt.