Seeing that Defense Against the Dark Arts is a crucial subject, both in Harry's life as well as a part of the school-going wizard's curriculum, it must be in the hands of a good professor.
Is it known who Minerva McGonagall appointed as DADA prof. once she was made Headmistress. Was it someone we know or was it just a random powerful wizard. Is there any information about this available?
The books and most other canon sources have bupkiss on this point. All we know for certain is that McGonagall is headmistress as of the end of the "present-day" (pre-epilogue) plotline, and that it's most likely not Neville (who eventually becomes the Herbology prof) nor Kingsley (who becomes Minister of Magic) nor HH&R (who are all left on the platform at the end of the epilogue; if they were professors of any subject they'd be on that train, or already at Hogwarts).
Most likely, the position is held by a former Auror, someone close to or in the Order and thus has ties to the school and to McGonagall; there are a few of them introduced through canon who (I think) are still alive at the end of the battle. Emmeline Vance comes to mind as one of those random Aurors (she was part of the advance guard that took Harry to Grimmauld Place in book 5), but I can't say for certain if she made it (EDIT: As per N. Unnikrishnan's comment, Emmeline Vance was killed prior to the events of HBP, so definitely not her).
Searching the book and Accio Quote shows zip.
McGonagall probably didn't appoint him, since as per Wiki:
As is confirmed in The Tales of Beedle the Bard, McGonagall becomes Headmistress of Hogwarts after the Battle of Hogwarts and Snape's death; however, in an interview Rowling said McGonagall would be retired by the time of the Deathly Hallows epilogue (19 years after the main events of Deathly Hallows), as she is "getting on in years."
The closest we have (from that same set of interviews) is:
Also, there is a new Defense Against the Dark Arts teacher, and that position is now as safe as the other teaching posts at Hogwarts, since Voldemort's death broke the jinx that kept a Defense Against the Dark Arts professor from remaining for more than a year.
J.K. didn't clarify whether Harry, Ron and Hermione ever return to school to finish their seventh year, but she did say she could see Harry making an appearance every now and again to give the "odd talk" on Defense Against the Dark Arts.
My guess is either Neville Longbottom or Pomona Sprout. They can't both be Herbology professors, and likely both of them are qualified to teach either Herbology or Defense Against the Dark Arts. They must have split in some way, possibly both of them teaching half of either class.
Of course, it's possible that Professor Sprout would have retired by that time, but I hope she hasn't.
The sad part is, had Remus Lupin survived, he would have made an awesome DADA teacher again, as parents would be more likely to accept him as a professor after the War, having seen that there are much worse things a Hogwarts professor can be than a werewolf.
The real question, though, is who would teach Potions? It seems there are very few people qualified for that position, Professor Slughorn would be retired again, and Professor Snape has died.
