I've been taking my recently turned 10 year old to the ROH for a few years now and we've generally had pretty reasonable views from the Amphitheatre for the price. I've looked back over where we've sat and we've had decent views from any of the following areas , with the caveat that she's needed a booster cushion (not only is she only 10 but she's on the small side). If you want a low priced seat in a central position then you will be looking at around Rows R and S upwards. The seats are pretty steeply banked and we've generally found that, with me occasionally swapping seats with her to avoid a tall person in front (and a booster cushion) the view is fine, especially for a ballet which is more of a spectacle and where the story is well known, such as Sleeping Beauty (and thus no reliant on being able to clearly make out facial expressions). We recently saw Romeo and Juliet from seats S60 and 61 and then from S54 and 55 the following week, and the view was fine.
We sat in T60 and T61 for Sleeping Beauty in 2017 (when she was 7) and she sat on my knee for the first act as she was struggling to see and we weren't aware of the booster cushions. A very kindly audience member passed us one for the second act and all was fine from then on. Again, not bad seats for the price and for a 10 year old the view should be okay.
We also liked seats R 74, 75 and 76 and there was no one behind which meant that, if my daughter had had problems with the view, she could have knelt on her seat. It's difficult to make out expressions from here particularly (which was a bit of a problem when we sat here for Winter Dreams which is so emotion driven) but it was possible to get full enjoyment from the other two ballets in that bill.
My current favourite cheap seats are towards the side of the Amphitheatre as it is possible to sit lower down (and thus get better views of the dancers' faces) and the price (for Romeo and Juliet which is probably a reasonable guideline for the Sleeping Beauty prices) is lower than for the central section of the Amphitheatre. You can sometimes find some of the action to the far side of the stage cut off but I think the view/price ratio is great. I have seen Romeo and Juliet from both seats D38 and D37 and was able to see pretty much all expression from here. Again, with a booster cushion a child would be fine.
I've found my children were so excited about going to the ROH that actually seeing the ballet was only part of the overall thrill! In addition, I've always found the staff unfailingly lovely with children.
I hope this helps.