Cook the bird until the breast meat reaches 157 degrees (recommended final cook temp by ThermoWorks). The temp will continue to climb while resting to 165 degrees. If you cook it until the breast meat reaches 165 degrees, it will climb to 175 degrees while resting. The breast meat will then be dry.
Rest your turkey after taking it off the grill for at least 30-45 minutes prior to carving.
I cook a turkey on the Weber every Thanksgiving and several more during the rest of the year. I use to cook them in a roasting pan on the cooking grate indirect. The last few years I have spun them on the Weber using a rotisserie (thanks to urging and prodding by
@Travis ). The weather in California around Thanksgiving is usually in the 50's. I start out 25 lit briquettes in each charcoal basket. I add 9 unlit briquettes per side every hour. A 14lb bird will usually take about 2 hours on a Weber. Figure about 11 minutes per pound in computing cook time. Make sure your bird is totally thawed and remove it from the refrigerator 2 hours prior to cooking.
Also, arrange your charcoal baskets more to one end.
When you place your turkey on the grill, make sure the legs and thighs are closer to the charcoal baskets. This allows the breast to be slightly further from the heat source and helps keep it from overcooking and drying out. If you want some smoke flavor, add some apple wood chips or small chunks at the very beginning. Using to much wood can turn the skin very dark. I like the skin to look like this
as opposed to this. I went overboard on the apple wood chunks on this one
I don't stuff the bird when cooking on the Weber. I save the drippings from the roasting pan and separate the fat from the usable drippings. My wife cooks the stuffing in the oven and then I pour the drippings over the stuffing, mixing it in when it's cooked. Stuffing the bird will extend your cook times extensively.