My favorite herbal drinks in the summer always involve Bo He (Menthae Herba, mint leaves), and to your question about BBQ, I would combine Bo He with Shan Zha (Crataegi Fructus, hawthorn fruits). Not only is the combination a good balance between the cool Bo He and the warm Shan Zha, but it is a blending of pungent and fresh minty flavor with the sweet and sour taste of Shan Zha. Furthermore, Shan Zha, which is a digestive herb with powerful actions to break down fat and protein, helps to counter the rich and heavy foods that might be in your BBQ. Bo He, which is a cooling diaphoretic, also cools and soothes the liver, and is therefore a good herb in warm weather to prevent the liver from overheating.
Cook the Shan Zha first for 30 minutes and then add the Bo He to steep. Proportions are completely up to your taste; I would try three parts Shan Zha to one part Bo He. Serve chilled but not ice cold.
If your BBQ includes beer and wine, another great herb is Ge Hua (Puerariae Flos, kudzu flower), which has been shown to speed the metabolism of alcohol in the body and reduce the symptoms of a hangover. I can’t vouch for the flavor of this one, though!