In "Games at Twilight," what key details suggest that the children use this rhyming game to choose who will be "it".

The best evidence that I can point you toward is the immediacy that the children fall into place and begin playing the rhyming game.  
The children decide that the first game that they all should play is hide-and-seek.  Then there is an argument about who should be it.  If you have ever played the game, you are familiar with this argument.  Mira quickly steps in to intervene, and she starts pushing and shoving kids into a circle of sorts.  While she is doing that, she is yelling "Make a circle, make a circle!'' Once the kids are in that circle, nobody explains anything.  There is no explanation about how the rhyming game works.  The rule about how and when to clap in rhythm is never stated.  Mira simply shouts out "Now clap!"  The children all know exactly what to do, and the children all know exactly when they have been eliminated and are safe from being "it."  That familiarity tells readers that the clapping game to choose a person is something that all of the children know and are used to.  

. . . and every now and then one or the other saw he was safe by the way his hands fell at the crucial moment—palm on palm, or back of hand on palm—and dropped out of the circle with a yell and a jump of relief and jubilation.