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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Closed Drafting: Closed Drafting (or ‘Card Drafting’) is a means of distributing cards or other game elements to players through an ordered, closed selection process – typically “select and pass”, sometimes called “pick and pass”. A typical implementation involves each player being dealt the same number of cards. Players then select one (or more) card(s) to keep, and pass the rest to their left. This continues until all cards are selected or discarded. 7 Wonders implements this type of draft. An alternative is that only one player is dealt cards, and they take one and pass it until all players have cards. This obviously is strongly biased towards the first player, and needs to be supported thematically and balance-wise. Citadels implements this type of draft.

Contracts: Players fulfill Contracts to earn rewards. These take the form of special goals requiring coordination and planning beyond simply being “first past the post”. These can be either public, where all players compete to be the first to complete them, or private, where only the owning player may fulfill them.

This often takes the form of a special type of Set Collection, such as in The Voyages of Marco Polo where Contracts defining an ad hoc set of resources may be claimed to obtain the corresponding reward. Pick-up and Deliver is a special case of Contracts common to train games such as Empire Builder which should be selected for those games.

I Cut, You Choose: A method of drafting where one player divides a set of resources into different groups; other players have first choice of which group to select.

San Marco, New York Slice, and Berrymandering implement versions of this mechanism.

This mechanism can also be used to divide more abstract resources, like turn order. Many games use the so-called “pie rule” or “swap rule” to minimize first player advantage, e.g. Hex. In this game, the first player makes the first move of the game, and then the second player can choose to switch sides (taking over the first player’s color as well as that first move), or to let the first player stay with that color and first move, and the second player then makes their first move in response to the first player’s move. After that, turns continue as usual.

Open Drafting: Open Drafting is used in games in which players pick (or purchase) cards (or tiles, resources, dice, etc) from a common pool, to gain some advantage or to assemble collections that are used to meet objectives within the game.

Saint Petersburg is a well-known game utilizing an open draft.
Azul utilizes a more complex draft, where tiles are selected from multiple constructed lots, with some reverting to the common pool.

Games where cards are simply drawn from a pile are not drafting games; drafting implies that players have some sort of choice and the ability to draft a card another player may want thereby denying them something they may have wanted. In Ticket to Ride, for instance, players may choose to draw cards from open display, so although most cards in a game are only drawn randomly, it still can be considered a drafting game.

This is distinct from Closed Drafting, the mechanic of “select and pass”.

Pick-up and Deliver: This mechanism usually requires players to pick up an item or good at one location on the playing board and bring it to another location on the playing board. Initial placement of the item can be either predetermined or random. The delivery of the good usually gives the player money to do more actions with. In most cases, there is a game rule or another mechanic that determines where the item needs to go.

Empire Builder is a classic pick-up-and-deliver game from 1982 that remains popular today. In this game, players build railroads between cities, and move trains around on the track. Players hold contracts specifying that specific cities demand specific types of goods. To fulfill a contract, a player must travel to a city where that good is available, pick it up, and deliver it to the destination city. When the player completes the contract, the player receives money as specified on the contract.

Set Collection: The value of items is dependent on being part of a set; for example, scoring according to groups of a certain quantity or variety.

Demonstrations:
A set of 3 of a particular item is worth points (but each item alone has no value).
A set of 3 different items is worth points (but each item alone has no value).
A particular item is worth 3 points, a set of 2 such items is worth 7 points, and a set of 3 of it is worth 13 points.

Example games:
Players collect and harvest different types of beans in Bohnanza, and they collect Monuments in Ra.

Contracts should be indicated when applicable, as a more sophisticated case where the items are defined by variable goals within the game which can be claimed, typically by only one player.