Is it a good idea to take a break from studying a mathematics PhD and revisit academia at a later stage?
Is it a good idea to take a break from studies if wanting to continue them at a later date, especially for mathematics? 
  


That's a tricky question.  There are probably some people for whom this is a good idea, but I wouldn't say it's generally one.  If it's possible in your current situation to essentially hit the "pause button" on your studies and be guaranteed the ability to come back (I think this is possible at many US universities), it might not be so bad, but otherwise you're taking a pretty big risk.  Also, in even as short a time as 6 months or a year, you risk forgetting a lot.  Certainly this is something a lot of other people I know who've taken breaks (say between undergraduate and graduate degrees) have mentioned. 

I would think about trying to arrange a shorter vacation (maybe 1 month) first, and see how that feels.  Taking a whole year is a genie that will be hard to get back in the bottle, and you don't know how you will feel, say, two months in.


  Would it hurt any opportunities later on down the line to take up a PhD a year or so from now?


It certainly might.


  Would my department or supervisor think badly of me for wanting to terminate my studies - and (generally speaking) would I not have to pay back the studentship instalments that I've been paid so far? (I've been funded solely by the University, not by a funding body such as the EPSRC).


I can't speak to them (or speak about the financial aspects), but probably it will not look great.  You should be prepared for everyone in the department to assume this means you are dropping out, no matter what you say about coming back.


  What would be some worthwhile things to consider doing during a break from academia?
  Has anyone else on here had experiences of a PhD (particularly maths) not going too well and feeling like they have had to stop or recharge their batteries?


I do know of one situation where this worked out a very well.  A friend of mine in grad school took a year off (after 3 years in the program) to teach in an inner city school.  I think at the time he started he was not so sure he wanted to finish, but the experience motivated him very effectively to go back and complete his degree, since he realized teaching middle school was definitely not for him.