If I may, I'd like to add a couple of points to Volume's very informative post.
Fatigue is a big player in pressure vessels. The infamous Comet MK.1 failures back in the 1950s were due to fatigue caused by cyclic stressing of the pressure cabin. Due to that, and over the years since, much has been learned about the fatigue of metal pressure cabins - we'd all be in serious trouble if it hadn't. But, the level of knowledge of composite fatigue, which happens in very different ways to metal fatigue, is much lower. Because of this, and the fact that one thing we do understand is that composite fatigue involves gradual weakening of the whole material (rather than crack propogation in descrete spots) it's necessary to apply large safety factors to any composite primary structure, beyond that necessary for similar alumiunium alloy structure. This forces the weight up, almost certainly beyond what would be shown strictly necessary if we understood things a lot better.
The second is that in it's ability to create complex shapes in single parts, the designer is capable of massively reducing the number of parts in an aircraft - the logistics and manufacturing cost reduction due to this is very significant. I recall a conversation I had with the Quality Manager at a certain aircraft manufacturer who was enthusing that for a large door assembly he could reduce the number of discrete parts from about 14,000 to about 300. This is a big pay-off for the cost of tooling up to a completely different type of technology.
What specific problems does Boeing have to solve? Well in my opinion the biggie is in the science of predicting the strength of a composite structure at any point in its life. If they can get to a really precise prediction of that, then they can keep the initial weight (and materials cost, although that's a lesser player) down to an absolute minimum and start to get the gains they're claiming.
G