Some countries have official, legal national sports. Others simply have unofficial sports that people say are the national sport. For both groups, I would say the criteria are a combination of these factors:
- The sport should have been invented in the country. Example: lacrosse
- The sport should be extremely popular. Example: kabaddi in
- If it's not popular, the sport should be "important" in some cultural
way and therefore worthy of conservation, reenactment, and reverence. Example: Charreada in Mexico.
- When possible, the sport should convey something meaningful about the country, ideally in a kick-ass way. Example: Varzesh-e Bastani in Iran.
For unofficial sports, current popularity would be a bigger factor than for official sports.
I wrote more about this topic here.