..

So prerequisites - low maintenance,

Lots of things depending on what you mean by maintenance.

suitable for kids to play on,

No can do. Grass is the most durable and even that will not survive the
attentions of energetic kids unless you help it along. Other ground covers
are less durable and will result in big maintenance or having to exclude kids
or both.

grows in
very shady area during summer, (but tree loses leaves in winter so has to
still survive), ok for humid conditions, drought tolerant (or happy with
grey water every now and then), doesn't cost too much to establish (very
limited budget), and if it was a mozzie deterant that would absolutely
wonderful as we have a problem with them as well when we're outside.


You are asking a lot here. As well as mozzie deterent why not ask for ant and
spider repellent? :-)

It doesn't matter if it takes a while to spread, I'm not in a hurry, but I
would like to do something - and if it can be grown from seed that would be
nice and cheap too.


Taking a while to spread is inconsistent with durability. One way that grass
survives wear is to quickly spread back into areas where damage has happened.

Thanks for any suggestions


Hard mulch (eg pinebark) or astroturf. No living thing will do all you want.
I am no fan of lawns (when my kids grew up the lawn went) but my experience is
that it's the only living playing surface that has a chance and that will take
some work.

David