There is a specific kind of boredom where you have access to thousands of games and cannot bring yourself to start any of them. It is not a shortage of options. It is that evaluating options is work, and you are bored precisely because you do not want to do work.
The rule
Let something else choose, then give the result two minutes before you judge it. That is it.
Two minutes is not arbitrary. Most browser games show you everything they have in the first ninety seconds — the core loop, the art, whether the controls feel right. If it has not landed by then, it probably will not, and the next press costs nothing.
Why removing the choice helps
A grid of two hundred thumbnails asks you to compare two hundred things against a preference you have not formed yet. A random pick asks you one much easier question: is this fun, yes or no. You can answer that in two minutes, and you will answer it about far more games than you would ever have clicked deliberately.
The side effect is that you end up playing things you would never have chosen. That is usually where the good ones are.