This is an expression from a very short letter. I quote it in full, omitting greeting and conclusion: 'My son tells me you have had a bit of a scare and I wanted to send you my very best and good wishes for speedy recovery. I will not send Nat down on Friday, but come myself, if I may.' I have no valid assumption of what had happened to the addressee. I guess that she was not well and her condition scared her. Looks probable?
Combined with "speedy recovery", I would say that it was a health issue of some kind, anything from an unexplained bout of fainting to a mild heart attack. In this kind of context and without further information I would think that it was a frightening incident but not a permanent or terminal condition.
I would not think it was simply a case of someone being frightened by something or someone.
Part of the problem in understanding your question, I imagine, was changing "you've had" to "you've got". They probably look similar but they often mean very different things. "... you've had a bit of a scare..." sounds perfectly normal (meaning "you have experienced something frightening"). "... you've got a bit of a scare..." doesn't make much sense to me. It would mean something like: "You are in possession of a scare."
It seems odd that the nature of "the scare" is not revealed in the book; are you sure it is not? "A bit of a scare" is a fairly common phrase, which is meant quite literally: that some event occurred which frightened the speaker. It may indeed be to do with the person's health, but it could be any unpleasant event. It may mean that, although the person was frightened by it, the consequences were less dire than expected (hence "a bit of" a scare).
I agree with the Mole, although I wonder whether the word "recovery" implies necessarily health matters... Can one recover from a shock or a rude word ? I know that"s what the Mole says, but I still wonder, in spite of her very cute avatar (is it from "The Wind in the Willows ?).
In this context I presume we are talking about parents old enough to be among that section of the population who are increasingly vulnerable to life-threatening health events. For them, to say that "I hear you have had a bit of a scare" suggests that this person either had, or was suspected to have, one of those nasty things like cancer, heart problems, or similar.
I am wrong again. It is "you have had a bit of a scare", not "you have got a bit of a scare". I am really sorry. In the text of the novel there is neither indications to the woman's health problems nor to the age she received this letter at. Anyway she could worry about her children's health. Thank you, JamesM, MatchingMole, Gregoire Samsa, Panjandrum and Ewie!