Q:

What impact has Douglas Hofstadter's superrationality had in terms of philosophy?

Superrational decision making is a type of rational decision making in which the players cooperate in a one-shot prisoner's dilemma without coordination, punishment, or magical thinking.
The idea is that when playing a symmetric prisoner's dilemma, one assumes that there is a unique solution to the mathematical problem of the optimal strategy, that this solution will be found and played by all superrational players, and that assuming the players are perfectly correlated, you maximize your utility.
The result in a one shot prisoner's dilemma is that two superrational players cooperate with each other, as opposed to two Nash-rational (or economically rational) players who defect.
A superrational player playing a Nash-rational economist will defect, and in general, in the absence of other superrational players, will play according to the Nash-rational strategy. It is only when there is a community of superrational players that one finds new types of rational behavior.
I have two closely related questions about the literature on this:

Douglas Hofstadter expounds this idea at great length in a series of Scientific American articles, reprinted in his collection: "Metamagical Themas", one of which is "Dilemmas for Superrational Thinkers, Leading Up to a Luring Lottery" (Scientific American, June 1983). I believe the idea, at least in its mathematically precise form, is original to him, and I credit him whenever I mention it.

Is the mathematically precise definition of superrationality in symmetric multi-player games due to him, or was it somewhere in the literature before?

Do philosophers take this idea seriously? I have not seen any professional literature which uses this. I am not asking whether philosophers should take the idea seriously, because I think they should. I am asking whether they do and if anyone can point me to specific examples of this that can be found in the literature.

A:

TL;DR:

Is the mathematically precise definition of superrationality in
  symmetric multi-player games due to him, or was it somewhere in the
  literature before?

Not really.  The idea had emerged several times in philosophy, economics and mathematics before Hofstadter wrote about it.  Notably, Martin Gardener wrote about a puzzle involving the notion in Scientific American in the 70s, in the same column that Hofstadter ultimately took over and wrote about superrationality in.
A good place to start would be SEP's entry on Common Knowledge.

Do philosophers take this idea seriously?

Yes.

Here's a quote from Hofstadter's Metamagical Themas (1985) where he provides a definition of superrationality:
You need to depend not just on their being rational, but on their depending on everyone else to be rational, and on their depending on everyone to depend on everyone to be rational - and so on. A group of reasoners in this relationship to each other I call superrational. Superrational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers. (Chapter 30)
Prior to this, the idea appeared a number of times, dating back to Hume.  See the linked SEP article for a discussion.
I will present two mentions from the economics literature.
In 2005, Thomas Schelling and Robert Aumann shared the Nobel prize in economics for "having enhanced our understanding of conflict and cooperation through game-theory analysis" (see the press release).
Schelling in particular can be credited with preempting Hofstadter's definition of superrationality in coordination games:

Thomas Schelling: There's a nice quote from The Strategy of Conflict (1960) that the linked SEP article uses: 
When a man loses his wife in a department store without any prior understanding on where to meet if they get separated, the chances are good that they will find each other. It is likely that each will think of some obvious place to meet, so obvious that each will be sure that it is "obvious" to both of them. One does not simply predict where the other will go, which is wherever the first predicts the second to predict the first to go, and so ad infinitum. Not "What would I do if I were she?" but "What would I do if I were she wondering what she would do if she were wondering what I would do if I were she … ?" (p. 54)
Schelling ran a variety of experiments based around games like the one sketched above, and ultimately developed the idea of a focal point equilibrium:
Most situations - perhaps every situation for people who are practiced in this kind of game - provide some clue for coordinating behavior, some focal point for each person's expectation of what the other expects him to expect to be expected to do. (p. 57).
Robert Aumann is the first person credited with a rigorous notion of common knowledge.  He presents the graph reachability notion of common knowledge on partitions on information space, which is now the defacto defition for modern logicians working on Epistemic Logic.  Here's the abstract of his paper Agreeing to Disagree (1976) (full text):
Two people, 1 and 2, are said to have common knowledge of an event E if both know it, 1 knows that 2 knows it, 2 knows that 1 knows it, 1 knows that 2 knows that 1 knows it, and so on.
Theorem: If two people have the same priors, and their posterios for an event A are common knowledge, then these posteriors are equal

A:

If your action forces other players to behave the same way, you are not playing a real game. you are playing a one-player game, a decision problem. The game theoretic fallacy Hofstadter makes is not new. An extensive discussion of the "symmetry fallacy" can be found in Ken Binmore's Game Theory and the Social Contract, Vol. 1: Playing Fair in Chapter 3.

