The following is a list of all board game mechanics I know in this category, and that aren’t too niche. Card game mechanics are also included. I’m posting this mainly for my own reference.

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Betting and Bluffing: Players commit a stake of currency or resources to purchase a chance of winning everyone’s stake, based on some random outcome like being dealt a superior set of cards or rolling a higher number. Players typically have partial information about the overall game state, and may “bluff,” by representing through their in-game actions that they hold a stronger position than they do. Conversely, players may “fold,” or quit the contest, and limit their losses to whatever they had already staked.

Dice Rolling: “Dice Rolling” is a game mechanism that can be used for many things, randomness being the most obvious. In wargames, “Dice Rolling” is used in conjunction with a variety of tables, notably a “Combat Results Table” (either as a Ratio or a Differential) which produces a result by strengths of both sides and a die roll (1D6, 2D6, and 1D10 are most commonly used). “Dice Rolling” can be a game in and of itself; see Yahtzee or Craps.

Die Icon Resolution: The player rolls a number of custom dice to resolve an event or conflict. Results must match specific symbols for success. This is not the same as Worker Placement with Dice Workers, as used for example in Roll for the Galaxy, where any result may be usable to accomplish actions selected after rolling.

Push Your Luck: Players must decide between settling for existing gains, or risking them all for further rewards, in a game with some amount output randomness or luck. Push-Your-Luck is also known as press-your-luck.

Here’s a description of the category by Bruno Faidutti:

“Double or Quits, Keep going or stop, cash your gains or bet them. The idea is not new. Many gambling games, such as Black Jack, make an intensive use of it, as well as some traditional dice game – and Pass the Pigs is only a modern version of those. This system is also used in many TV games, where the winner can either leave with his gains, either answer one more question at the risk of losing everything he won so far. Like in Luigi Comencini’s Scopone Scientifico, if you never stop, you always end losing. This system is also very efficient in board and cardgames, since it generates a high tension, and some anguish when rolling some more dice or drawing one more card. The best known “double or quits” game is probably Sid Sackson’s Can’t Stop, a clever dice game figuring, in the last edition, mountaineers so impatient to reach the summit that they usually end up falling down. It’s even trickier when all players are aboard the same ship, not knowing if, or when, it will sink. Time to leave or not? That’s what happen in Aaron Weissblum’s Cloud 9, as well as in Diamant, a game I designed together with Alan R. Moon.”

Here’s the introduction of the Jeopardy dice games from Reiner Knizia’s Dice Games Properly Explained:

“You focus on progessing and maximising your results. But the stakes are rising. If things go wrong, you lose it all. Great risks bring great rewards – or utter defeat!
Disaster strikes in many different ways. More than ever, you need to weigh up the potential gains and losses. Rolling specific numbers or reaching certain totals may catch you out. You see disaster looming – but can you escape? Other games allow you to continue throwing as long as you keep your options open. Know when to stop and secure your results. If you get greedy and your luck fails, you are out. You need to make the right decisions and be lucky, too.”

Random Production: Resources are generated from a random process and distributed to qualifying players. CATAN and Crude: The Oil Game are based on early examples of this mechanism.

Re-rolling and Locking: Dice may be rerolled, or may be locked, preventing rerolling. Yahtzee and Cosmic Wimpout are early examples of this mechanism. More recent games include the idea of “locking”, where certain dice are prevented from being re-rolled, as in Zombie Dice and Escape: The Curse of the Temple.