It is customary when declarer is in the dummy to say "on the board"? Where did this terminology come from?
The dummy's hand is typically arranged flat on the table (a flat surface, or "board"), and it appears the terminology comes from this
For example, from the ACBL's glossary:
BOARD. (1) A duplicate board. (2) The table on which the cards are played. (3) The dummy’s hand, so called because it lies on the table.
The question is not quite clear, but you may also be asking why players say "On the board". It is not polite for opponents, and not permitted for dummy, to comment on the way a hand is being played, but one of the few exceptions is to point out when declarer is about to play a card from his hand rather than from dummy as he should do. "Excuse me, but I believe the lead is in dummy rather than in your hand" is somewhat cumbersome after the first few occasions, so just saying "On the table" (or presumably "on the board" in some circles) is the normal phrasing.
In many card games, the "board" refers to exposed cards. The "dummy" plays this role in bridge. The declarer and both defenders can see it because the dummy's cards are placed face up on the table.
In Hold' em (poker), the "board" refers to the five exposed (common) cards that everyone can see and use. This is in contrast to the two private (hidden) cards in each person's hand.