My 7 year old wants to learn an instrument- which one?

Viola!

Just to give an idea of violin starter costs, dd got her first new 1/10th violin through her teacher for about £80, then the second one, 1/8 Primavera, was part exchange, we paid about £70 as dd scratched the first one. Then dh bought a spare 1/8 Stentor for £30 on ebay. It was used but in excellent condition. So, unless you are buying Gligas which don't depreciate :), it is pretty cheap.
But I think any instrument will be expensive in a long run taking into account that long run equals proficiency and that the instrument should match the level of playing.

Don't be fooled by the cheap violin starter sets - violins are not cheap in the long run. Brass and non-reed woodwind are cheaper as minimal maintenance costs and only need to upgrade when reach grade 5ish. Violins need changing for next size up as they grow and strings vary in price but the recommended sets for kids are around £30-50. Saying that your child maybe too small for most brass/ woodwind instruments at the moment ...

yy re loaned instruments - DS2 is also learning trumpet now and his instrument is loaned from the county - if he likes it and gets on with it we will buy one.

She's probably a bit young for wind instruments (embouchure needs to be developed ie teeth in) but my DD has played clarinet since about 8-9 yo and we love it.

DS2 has played piano since he was 7 so that's an option. He plays it in band at school as well tho I agree other instruments are more flexible (no band needs more than one piano but can usually make room for extra violins etc).

Violin and other strings are a bit grim IMHO until they are well played (this can take years judging from the many concerts I have attended!). I do love a cello tho.

Second hand pianos are cheaper (ours was about £1200) and you can often hire one from a music shop for not much.

Also, not sure where your DDs lessons will be but while my DD learned piano privately, she added lessons for bassoon via school via the County Music service. They often have schemes for loaning instruments cheaply or even free in some cases; that's how my DD got started. They may also have a 'try out' day when your DD could try a range of instruments she hasn't thought of. (Now is a good time of year as people get ready for next year's start in September.) It is definitely true that people 'fit' different instruments. I had no idea what my DD's teacher meant when she told me that my DD was a 'good fit' for a bassoon but several years on, I know just what she means! Not sure how it works county to county but maybe Google your own county with County Music Service and see what comes up?

All great advice. All I would add is perhaps learn an instrument that is being tuaght in the school by a peripetic teacher. I.e not just the recorder group at lunchtime, but maybe a visiting teacher comes regularly and runs a violin or guitar group etc...
This way you don't have the added stress of getting anywhere else for the lesson, and the group mentality/friendly competition seems great for motivation...
Another thing to consider is does your Local Authority have a music hub? i.e my kids have started on their instruments and learnt enough after a couple of terms to join a Saturday group where others are learning their instruments and again that is really motivating.
Finally, some instruments can be bought fairly inexpensively which might even work out better value than renting them...

I learnt piano as a child and hated it. I found reading and playing two lines of music hard and it was boring doing it on my own. But I love music and as an adult took up the clarinet. Much easier (one line of music only) and, as others have said, the social aspect of playing with others is so much more appealing. Piano may be a good place to start for a bit (most decent musicians seem to be able to play piano at least a bit) but be prepared to consider other instruments as well / instead later.

Something with fucking headphones. I'm not kidding. If I have to hear that fucking violin scrape up and down scales for one more fucking moment I swear I will lose the fucking run of myself. Its agony. You've been warned!

I am not musical at all. Well... I learned piano for few months and then dropped, so it does't not count. Three years on since my dd started music lessons, I feel quite professional on a subject , so much so that my piano teacher friend asks me for advise on books, events, and other info :)) It is a great fun and I highly recommend music lessons.

I think violin hurts, yes, but depends how quickly things are going, depends on a teacher too. It took dd few months to get used to the string pain

A kid who wants to learn an instrument off their own bat is more likely to do well and enjoy it than a child who's there because their parents think it's the right thing. Your daughter sounds ace.

Just seen your post asking if it matters about your lack of ability ... my DD outgrew my help after about four lessons when she was six! I have no idea about music, spent years sitting in the audience learning what all the instruments in the orchestra are. To this day, (she's 15 now, grade 7 on two instruments) Music Mums will talk to me in a way that assumes she must have inherited her musical ability from me, and I have to smile and say no, it's all her own. She has had a great teacher, though, so has someone to talk to about it all, which I think is important.

Has she thought about wind instruments as well? My DD started on piano at six, and I agree with the comments above about the various benefits. My DD had a keyboard on a stand at first to practice with - cheap and portable option till you're sure she will stick with it. BUT you can't join an orchestra with a piano - other instruments would allow her to play with others which is a huge part of the fun in learning. My DD moved onto wind instruments later on - there are some things which are better learned as second instruments when you already know the basics. Also, sounds obvious but consider the cost of an instrument and how portable it is ... I didn't ....!

Agree, violins are relatively cheap, and portable. Steer clear of anything brass as they are a small fortune. If DD shows musical talent and sticks with her practising them you can add piano later, by which point she will have a grasp of musical theory which will speed up her progress. Then you can remortgage and buy a piano.

Piano is a great basis for anything else, and doesn't require the physical development of most other instruments.

You can always pick up a touch sensitive electronic keyboard for practice in the early days - far from ideal but better than nothing.

Cello is a brilliant instrument in my book.

DS is 3 and stoutly maintains he wants to play double bass. I've told him he'll have to wait a bit.

I would recommend piano! :)
Pros Children like piano for whatever reason, you can see results very quickly-you just press the key and get a nice sound (unlike with strings) and it is good from point of view of musical basics, music theory in a future, etc.
Cons It is expensive and it is a solo instrument

Second choice would violin - it is light and easy to carry, it is not expensive, it is an orchestra instrument and therefore very social, and when she is grown up she can always play it for hobby in some local orchestras, at least.