What does the character 棋 mean in Chinese board game names?
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Direct answer
棋 (qí) is the generic word for a board game played with pieces, not a name specific to any one game. It is the second syllable in 象棋 (xiangqi), 圍棋 (weiqi, go) and other board games, the same way "chess" in English can prefix onto "Chinese chess" or "shogi is Japanese chess" without those games being related in rules.
What the first character is doing in each name
It is the part that actually distinguishes the games: 象 (elephant, or possibly image, see why it's called elephant chess) for xiangqi, 圍 (surrounding) for weiqi, describing go's core mechanic of surrounding territory. The shared 棋 tells you only that both are board games with pieces, nothing about how they are played.
Why this matters for reading Chinese sources
A text mentioning "棋" in isolation could be talking about any board game of the era, which is part of why early references to games under various names cannot automatically be assumed to describe modern xiangqi specifically. The generic character is exactly why name continuity is not game continuity.
Related questions
Is 棋 used for western chess too?
Yes, western chess is commonly rendered 國際象棋, literally "international elephant game," borrowing xiangqi's own name as the base term.
Does 棋 ever mean just "a single piece"?
In some compounds it can refer to a playing piece rather than the game as a whole; context decides which sense is meant.
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