The most common and most likely answer to this crossword clue is the 5 letter word HALMA
I've seen this crossword clue in Independent
H
A
L
M
A
Buddy explains!

The straight part
We know that "Game on board" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case game on board = halma.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HALMA mean?
HALMA noun- A board game invented by George Howard Monks in which the players' men jump over those in adjacent squares.
- In the Greek pentathlon, the long jump with weights in the hands.