I would propose a multi-sport of sports which are indigenous and popular to a country or region and have potential to have mass popularity like Kabaddi from Indian subcontinent, Australian rules football, hurling, sepaktakraw etc.
Since most of them are team sports, the quotas can be handed as follows - four best teams and one each from Asia/Oceania, Africa, Europe and Americas; so each sport shall feature 8 NOCs.
NOCs may not be ranked by medal tally but by cumulative points earned in sport. The points earned shall be earned by combination of final standing and a weightage factor decided by whether the sport is indigenous in that region. For example, in hurling Ireland shall have a lower weightage factor then, lets say Kenya.
Probable List of sports
American football
Basque pelota
Hurling
Kabaddi
Sepaktakraw
Wushu
Sambo
Netball
Lawn bowls
Australian rules football