I'm not sure how general the expression is but, where I come from, if a new-born baby shows unusual interest in its surroundings (for example, looking around for its car keys), then the Wise Women nod and say, "She's been here before..." It's been a while since we had an interesting word, and you could say that this is an example of anamnesis. Or, at least, an example of the most interesting variety of it, as it's a word with curiously different meanings.
At root, it's just a Greek word simply meaning "remembering" or "to call to mind," and is used as the name of that figure of speech, where the speaker brings to the audience's mind things from the past ("Do you remember the days when bankers were a byword for probity?") But it has some more technical uses.
In Christian theology, the word refers to the "remembering" part of the Eucharist -- "This do in remembrance of me". Theology is an incredibly well-analysed and labelled field of study, which can be a goldmine of useful words for those half-grasped concepts you feel ought to be new but were, in fact, thoroughly turned over by scholastics in the Middle Ages; this is not one of the more useful or entertaining ones, though. [ A note from our Well I Never! Department: Did you know that "total depravity" is not just a clichéd expression for "rather naughty", but a quite precise theological term? Well I never!]
The most interesting use of anamnesis is, I think, in the philosophy of Plato, where it is Socrates' answer to the question: "Sir, Sir! If what you are looking for is really new and unknown, Sir, how will you recognise it when you find it, Sir? Well, Sir, how can you, Sir?" Socrates, of course, being Plato's glove puppet perfect teacher, does not react to such annoying interruptions to his chalk'n'talk by throwing the blackboard rubber with deadly accuracy*.
Without going into philosophical questions which I am unqualified to pursue, the view of Socrates and/or Plato seems to boil down to: We have all been here before, and what we call "learning" is really a process of remembering what we have forgotten. That is, what we have forgotten about the truths we learned in previous lives, especially those truths recovered from the true, deep eternal knowledge of which contingent "real world" beliefs are a mere distorted reflection.
These are curiously attractive ideas, especially if you are of a conservative bent of mind. A lot of Western art and culture is "neoplatonist" in spirit, without necessarily knowing it. You could even go so far as to say the emphasis of traditional science on "discovery" versus "invention" (i.e. Newton is said to have discovered, rather than invented, gravity) is a form of anamnesis, and that post-modernism is/was the attempt of the disillusioned and sidelined Humanities to turn the tables. I don't think science has really noticed, though ("What's that funny little buzzing sound?").
There is yet another meaning to the word. Have you ever had that mildly annoying thing happen to you, when waiting to see a doctor in a hospital: a medical student sidles up and asks whether they can ask some questions while you're waiting? And it turns out they're practising taking a medical history of your ailment? Well, the technical term for that is anamnesis and, believe it or not, I discover that the mnenonic used by British medics for the process is SOCRATES ... (Site, Onset, Character, Radiation, Associations, Timing / duration, Exacerbating / alleviating factors, Severity). I wonder how many medics these days know quite how apt that is?
* I realise this portrayal of a classroom is at least 30 years out of date, not least in the idea that any child today would resort to something as subtle as a clever question in order to annoy or divert a teacher. Plus any teacher resorting to assault with a pedagogical weapon would quickly end up in court.