Children's version of Rummy?
Far be it for me to answer my own question, but...  I've play tested the partial rules from wikipedia with my children, and have worked out the following fuller set of rules, if it's helpful (also trying to make it more rummy-like):


For 2 players, each player is dealt 10 cards.  For >2 players, each player is dealt 7 cards;
Players immediately put down all cards of the same value (example, two 6's or three Kings) face up in piles in front of them;
Once everyone has put down their starting-hand matches, players take it in turn to put down cards matching other people's matches (this differs from the wikipedia rules: wikipedia said to do this extra laying off all at the same time - that becomes rather frantic with children as they argue who put down a card first if there is more than one match).  If they lay down a match, they take the pile to their part of the table;
Once all above starting play has stopped, the dealer hands out a new card to each player in turn, and the pile building and taking is repeated by each player in turn until either:

all cards have been dealt and no more matches have been played (most of the time, all cards would be matched by now, but sometimes they can't be, e.g. with 4 players each player could end up with one of the same card number);
a player manages to lay off all the cards from their hand (this is again different to wikipedia's rules, but makes the game shorter if you have young children playing - keeps their attention a bit better by allowing more rounds to be played);

Once either of the above conditions is met, the cards in front of each player are counted (count of number of cards, not face values).  The player with the most cards wins.  Optionally, you could subtract any cards left in players hands from their score if using the second of the above ending conditions.


My children picked up these rules pretty well and really enjoy playing it!