Are we talking about a problem question at university or some kind of assessment exercise for a Vac Scheme or Training Contract? If it's the former then I'd refer to them by his or her name, if it's the latter then I'd probably just say 'the client', but I don't think it really matters all that much.
Name, IMO, though it depends on the circumstances
For what it's worth if this is legal argument in the sense of written submissions it is my view that you should use claimant/defendant or appellant/respondent.
I would be smacked around the head by my pupil master if I used 'the client' in the course of written work.
Mr X might be appropriate in some circumstances, for example where writing an opinion before proceedings have been commenced.