In Rules of the Game, Waverly's mother says the "strongest wind cannot be seen." What does she mean by this? How does this idea show up in the story?

Waverly says she learned the art of invisible strength from her mother. The invisible strength of the strongest wind means she does not always say what she is thinking and she acts as a hidden threat. This is one of the ways she is able to become so successful at chess. She watches and learns. No one really expects a little girl in ribbons to be a chess champion.
Waverly was good at chess because she was intelligent enough to learn from what others taught her. Her mother realizes the value of Waverly becoming good at chess and allows her to play with the men in the park. Waverly watches them.

I learned why it is essential in the endgame to have foresight, a mathematical understanding of all possible moves, and patience; all weaknesses and advantages become evident to a strong adversary and are obscured to a tiring opponent. I discovered that for the whole game one must gather invisible strengths and see the endgame before the game begins.

The wind becomes a metaphor for Waverly's inner voice. She listens to her instincts, the wind inside of her, and she becomes more and more successful. She uses these insights to secretly sneak up on her opponents.

"Blow from the South," it murmured. "The wind leaves no trail." I saw a clear path, the traps to avoid. The crowd rustled. "Shhh! Shhh!" said the corners of the room. The wind blew stronger. "Throw sand from the East to distract him." The knight came forward ready for the sacrifice.

The concept of the strongest wind does not just apply to chess. Waverly uses her wind to give herself a voice. In a way, she was following her mother's own advice when she told off her mother. She just did not quite follow it in the way her mother intended. Waverly was just a little too rude. She was independent, but forgot to be strong and silent.