Seating Arrangement Problem

[tshort]

Schostic generation of 8 x 8 latin squares would be a "fair" method of determining seating.

Latin squares are n x n tables with n symbols such that no column or row repeates a symbol. So, they would satisify your criteria of each person starting in each position relative to the button. Any given 8 x 8 latin square wouldn't guarantee fairness in relative position to other players. Randomly generating the latin squares would be a fair method of seating (although it could no longer be "fair" after the it is determined). Building in constraints to the random generation could further guarantee fairness.