##Go for a more abstract name that invites users to consult the FAQ.
One thing that seems to be established, both by the data and the experience of posters here is that there is considerable confusion about the scope of the site. The primary evidence is the high closure/deletion rate for new questions.
What is disputed is the root cause of this confusion. @Rachel hypothesizes that the name/URL of the site is a contributing factor, and others have asked for evidence to support this.
I don't think it is possible, without any kind of budget, to tease out the reasons for this confusion. My intuition suggests that it is surprising that a site entitled "programmers" is in reality a lower scope. I also suspect that the history of programmers.SE, which started as non-programming-related.SE, a release valve for software-related questions that don't belong on Stack Overflow, plays a role in that. In fact, the founders even recently suggested the programmers.se be used in this way.
Changing the name from programmers.se would have the advantage of both moving away from a general term and making a break from this legacy. This would have the downside of some work and some confusion. My judgement is it's worth it.
Some commentators have noted that the most popular sites, StackOverflow, have even less specific names, yet do not suffer from this confusion, and therefore a name change is unnecessary. What I think this, as well as things like the Arqade rebranding, suggests is that if a site's scope can not be easily captured in two short words or less, it may be better to have a more more abstract name that invites users to consult the FAQ to get a more concrete idea of what the site is about. This may also help the site to form a unique brand.
What type of name might work for this community? Some have suggested things like "The Whiteboard" which doesn't excite me, but I don't have any better ideas. The celebration of Alan Turing this past week leads me to consider the possibility that we could use the name to honor those who have gone before us in software development.
In any instance, I think both the past track record of SE sites and the current future direction suggests this.