How long a ride would you consider?

Many of the Orange Line stations don't have too much around them. And many BBQ places may be open only at lunchtime.

If you ride as far as the southwestern medical district/parkland station, there are some good hole-in-the-wall places within a half mile or so: Maple & Motor for burgers, Avila's for Mexican.

My usual recommendation for BBQ near the airport is Hard Eight, but there's no public transit, and even though it's a 5 minute drive from the airport, a cab there has a $22 minimum airport fare.

A better bet might be the Grapevine shuttle. Picks up right at the Hyatt, runs till 10/11pm, $5 for a day pass. Ride it to Historic Downtown Grapevine, and there are lots of choices there.
http://www.grapevinetexasusa.com/shu...es-and-routes/
http://www.grapevinetexasusa.com/sho...dine/downtown/