Announcing that you've been backed by YC (or any investors) can be enough of a news story to generate some good press, which is most useful if you can control the timing to coincide with other interesting things that you've done (like big feature deployments, major partnership, traction announcement) or want to do (like raise money or get the PR ball rolling).
By naming themselves now, they'd be throwing away future exposure without receiving any immediate benefit.
I don't know what phase the mystery startup is at, but while you're very early stage it's also often desirable to enter investors' radar with a bang when you're ready to raise money, and not before, since novelty & freshness can help offset the lack of evidence young companies are defined by.
This is a good point, but the OP brings up a valid question that can be applied to businesses in general. The smallest companies stand to lose the most for exposing their idea, yet companies from 37Signals to Apple are very secretive about what they're making next. Why is that?
Your point applies to the small guys who haven't launched yet, but it's not applicable for established companies. Sure, the big guys are on everyone's radar and others will copy them if they release their idea, but they're big guys for a reason. They've built a brand that some copycat doesn't have. In addition to that, they stand to benefit the most from exposing their idea because they already have an established customer base from which to gather feedback.
The "it's not about the idea" idea refers to people with no technical background who think they can get rich simply by having an idea, like say, "a site where people can upload and watch videos".
By naming themselves now, they'd be throwing away future exposure without receiving any immediate benefit.
I don't know what phase the mystery startup is at, but while you're very early stage it's also often desirable to enter investors' radar with a bang when you're ready to raise money, and not before, since novelty & freshness can help offset the lack of evidence young companies are defined by.