There is an Oyster Card for children aged between 11 and 15, but you have to apply for it in advance and supply a photo. I think it's aimed at people who live in/regularly visit London:

https://tfl.gov.uk/fares/free-and-discounted-travel/11-15-zi...

The other option is a child's paper ticket - those can be purchased without having to apply for anything in advance.

Adult fares via Oyster/contactless may still be cheaper than a child's paper ticket.

You can use the Single fare finder to check the price of a specific journey - for both adult/child:

https://tfl.gov.uk/fares/find-fares/tube-and-rail-fares/sing...

If the Oyster/contactless adult fare is cheaper than a child's paper ticket, you could just use a standard Pay as you go Oyster card:

https://tfl.gov.uk/fares/how-to-pay-and-where-to-buy-tickets...