The illustration accompanying this article is a detail from Brueghel's "Children's Games" which was painted in 1560.
While it can't be certain what the children are doing in this picture. It's very probable that they're playing some variant of "Odds and Evens". One child holds a small number of objects; buttons, marbles, pennies, etc. and the other child has to guess whether the total is odd or even. A correct guess wins a token. A wrong guess loses one. Traditional Games of England, Scotland and Ireland states that the game dates back to "Ancient Greece and Rome".
They might also be playing "Eggs in the Bush". One player holds a number of marbles in one hand and the other players have to guess the number of marbles. Those who guess correctly are paid that number of marbles. Those who guess wrongly must pay the holder the difference between the number guessed and the number actually held. Players take turns holding marbles.
Another variant of this is "Prickey Sockey" which was played at least as far back as Victorian England. On New Year's Day, children beg for pins from their parents with the phrase "Please pay Nab's New Year's Gift". I found no reference to who "Nab" is. Children start the game by reciting the poem
A pin is concealed in the closed fist and the opponent has to guess which way the pin is facing by reciting "This for prickey." while pointing at the guessed point end, and "This for sockey." while pointing at the guessed head end. A correct guess wins the pin.
I tried to determine why this game was popular around New Year's Day. The only information I found that seemed relevant was from The Hat Pin Society of Great Britain (http://www.hatpinsociety.org.uk) about hat pins during Queen Victoria's time. The children may have been imitating their parents.
"Alarmed at the effect the imports had on the balance of trade, Parliament passed an Act restricting the sale of pins to two days a year, at the beginning of January. Ladies saved their money all year to be spent on pins in an early example of the 'January sales'! This is thought to be the source of the term 'pin money.'"
The Elliott Avedon Museum and Archive of Games
Gomme, Alice Traditional Games of England, Scotland and Ireland (London Thames and Hudson, 1894, 2 vol., ISBN 0-500-27316-2, $18.95)
The Hat Pin Society of Great Britain