I think the place would have to have a flight theme, similar to what Spirit Rest. near MIA has- waitresses dressed as stewardesses, first class seats taken from Eastern and Pan Am jets, etc. Not to mention the plethora of Eastern Airlines posters around. Since it would be like a Friday's, why not Flyday's. Bad pun, better something like the Mileage Run Cafe and announce "boarding times" when patrons have their named called when their table is ready.
As for reward scheme, I think you have it figured out upon number of visits combined with $$ spent plus throw in a double or triple points bonus on certain days or when paying with Amex for example.
Since we have a world of travelers, food would definitely have to be international and of course- special meals available upon request- vegan, kosher, etc.
Or building on Randy's idea- House of Miles. Good blues music and bbq, good- but that already exists as House of Blues. ;(
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"Fly me to the moon and let me earn alot of miles."