Thanks for all the feedback - lots of food for thought. Esp interesting to hear from shaggymaggy the music teacher's perspective. dd's music teacher is really nice and patient and seems good at making it 'fun' - but doesn't seem to understand how unmotivated she is!! He says she is very musical and just needs more involvement to enjoy it/do well - but maybe he is just saying that to be nice...
She chose clarinet aged 7 because an older friend did it - and I suspect because I suggested she play the piano, which we already owned and which she seemed to have a natural talent at - she was therefore determined in typically stubborn style to do what we hadn't recommended!
She has never really listened to clarinet music of her own volition, classical or jazz, both of which she's done at school (her current teacher suggested she switch to jazz clarinet, which was better initially). Whilst the friend who she copied has turned out to be a clarinet prodigy, grade 8 by age 13, etc - so probably a bit demoralising for dd who is still only at grade 3!
I think she wants to continue only out of a sense of not giving up and because she's already learned some of the grade 4 pieces. She also says she wants to work 'in music' (rather vague, teenage-y ambition) and has talked about doing music GCSE - would she need to keep on with the clarinet for this (or get another instrument up to a high grade v quick)?
She did play in the school wind band in year 7 but dropped out because 'the pieces they were doing were too hard' and I suspect she only wants to do lunctime/after-school activities if her friends are doing them too - none of them play a wind instrument. The music she actually enjoys listening to is all guitar bands, so clarinet doesn't really fit in there - if she could 'join a band' like her pop idols, she might enjoy that, but she'd need to learn the guitar/bass/keyboards or sing instead.
I think having the lessons at school has also been part of the problem - she has to miss normal lessons to get to them, so misses some academic work and not all teachers are helpful - some of them tell her off for being 'late' or absent even though she was at her clarinet lesson! The lesson times change and she's missed several lessons by getting the time wrong - I then get cross and she doesn't progress, but I think this factor does make her feel even less positive about the music lessons, as she's not the most naturally organised person. At her old school, she had problems with the music teachers too - they kept changing/were ill and missed loads of lessons etc and only 1 had clarinet as their main instrument - most did a bit of clarinet 'on the side' and none were very good teachers. Her current teacher IS good and does turn up for lessons - but think it may be too late to get her motivation back. I certainly can't 'nag' it back. 
Overall, I think giving up this term is the way to go (unless she needs to keep it up for GCSE music or is determined to pass grade 4). My parents - who are retired - pay for her lessons and it's not really fair on them to pay for something she makes so little effort at. But am pleased to hear of others whose dcs did later take instruments up again. So will keep hold of the clarinet and hope...
I do see a difference with dd2 - she's learning piano and I have never needed to nag her to practice - she's naturally probably less musical, but seems to really enjoy it. Which is how learning an instrument should be, surely?...