'Win the button' tournaments are like normal tournaments. Only difference is if you win the current hand, you will get the button next hand.
What happens when a tie occurs? Then there will not be one winner, but multiple.
Most tie breakers in tournament / live cash game format is the "left of the button" rule. Left of the button has most priority then it goes around until the button (least priority). This happens with these an extra dollar in a split pot or to determine who shows their hand first in an unraised pot... etc...
I assume the logic is that left of the button has the most disadvantage in a hand, thus with a tie, it gets the tie breaker. Just like in chess, a draw playing black is more valuable than a drawing playing white.
I don't have any experience with this format, so not an expert answer. But since it seems there is not a set precedence, consider using bridge suite to break ties, like the opener is determined in stud poker. You tie with AK for example whomever has the highest ace gets the button. the suite order highest-down is Spades, Hearts, Diamonds, than clubs, going low to high it is alphabetic order.