Paint sets

Rowney do small water colour sets, usually sold for adults. I suspect anything aimed at children will be poor quality.

I use pots of liquid paint with mine.

I remember, as a child, those awful metal tinned paints that always cut your fingers when yo tried to clean them up!

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Inexpensive tubes of acrylic paint...from those book remainder type shops.....and then get an old ice cube tray and squirt a little in a few segments. You will probably need 6 or so main colours and then have the fun of mixing others.
Acrylic paints are water soluble so can be mixed with water to be less gloopy, and brushes tools etc will come up nice and clean under the tap.

It might seem a bit more expensive initially but if you keep the tubes well sealed they will last ages. (you can always have a dabble yourself)

Yes, a couple of good ideas, there. Thank you.

Would the problem with acrylic paint not be that when they dry, and they dry very quickly, they are no longer water soluble, so they are really only usable while they are wet and liquid? It's the consistency of oil paint, but, as you say, water soluble, but with very fast drying qualities. You can buy reasonably priced student water colour tubes and, as HildaW says, squeeze them into an old ice cube tray and let them dry. They'll last for ages and the colour will be better than any child's paint box. The Works sell good, reasonably priced water colour tubes and sketching boxes (little square pans in a box).

acrylics dry no faster than old fashioned child's palette paint, in fact as you can use it in varying thicknesses it can take a fair bit longer to dry - it can be used in the style of both water colours and oil paints.
For young children I've used it as I would water colour, but the 'dry brush' technique rather than in washes.