My kids are younger and have taken to caving enthusiastically, but friends have had different experiences.
My main focus is warmth and comfort; if they are comfortable, they will be much more likely to enjoy themselves. Once they've got thd bug, they'll enjoy the miserable stuff too
Below is what I'd go for, for maximum comfort.
- Lots of layers for starters, or a wetsuit if it's going to be wet. I got one from the local car boot sale for a few pounds and there are loads for cheap on Facebook marketplace.
- Halfords do kids cycling gloves that are very good for caving.
- A helmet that's the right shape for a light, so either a climbing helmet (petzl picchu is good) or a skating type helmet is probably better than a bike helmet.
- A light that isn't too heavy and in an ideal world is balanced front to back. My choice is a Petzl Myo, but you can't buy those new any more and they are hard to find 2nd hand.
- Harness to help on climbs. I have an old Petzl Ouistiti and an Edelrid Fraggle. Both are good, but the Petlz is easier to clean as it is all tape and buckles (though a belt is probably better for a 9 year old)
- two pairs of socks!
- boiler suit for dry caves, rain suit for wet caves. I've also got a Warmbac oversuit in XS that a club member gave me, but they are really expensive, so I'd not bother buying a new one until you are sure she's keen.
- as Chris says, elbow pads can be used as kids knee pads. I wouldn't buy them specially, but if you've got your own anyway, might as well give them a try.
Then choose the right cave. Don't do anything too hard too soon. And as Mike says, bring snacks!
Make little challenges for them, that they can try several times and build confidence while having fun. I like to send them on a time trial around a very short loop, crawling under a boulder, then back out over the top of it. Last time, we got down to 17 seconds, from over two minutes at the first try.
If they aren't having fun, stop and tell a funny story about a similar thing happening to you. By the time you're done and they are laughing, the problem will be much easier.
Take some photos, so they can show off to friends and family, then they'll have a bit more emotional investment in it.
Taking a friend is sometimes great, but tears can also be contagious, so pick the right friend if you do.
Finally, have fun!