I'm cooking a full chicken divided into 8 pieces...
When is the best time to put sauce on chicken cooked on a BBQ?
Is it better to do it after it sits for a while on the BBQ? Is it better to let it sit in the sauce over night?
If you let it sit in the sauce overnight then it's no longer a sauce, it's a marinade.
That's fine, but marinating is something you generally do with tough, cheap, and/or dry cuts of meat such as chicken breasts or top rounds (beef). For a full chicken, especially the wings, drumsticks and thighs, a marinade is entirely unnecessary and in my humble opinion simply dilutes the deliciously moist interior and often makes the whole thing soggy and depressing.
Generally the goal (or at least my goal and that of the majority of restaurants I've been to) for chicken wings or even a traditional roasted whole chicken is a nice crispy, golden-brown, well-seasoned skin with juicy meat on the inside.
You can't get that if you douse it in sauce before grilling it. The water in the sauce will inhibit the Maillard reaction that causes browning (and crisping), and as Mike points out, the sauce may even burn. In fact, it almost certainly will burn; most store-bought BBQ sauces only last a few minutes exposed to direct heat.
With that in mind, the best time to sauce a grilled (BBQ) chicken is after it's cooked, or more precisely, when it's nearly cooked. Go ahead and season it liberally and coat it in butter or oil beforehand, but wait until it is no more than a few minutes away from being done before you sauce it. You're not trying to cook the sauce, and it takes no more than a few minutes to get the sauce to bind to the skin - i.e. a few coats with 30-60 seconds each to reduce and form a sort of glaze.
This is true for almost any meat and any cooking method as long as you're not marinating - BBQ chicken, fried or convection-baked wings, smoked or oven-braised ribs - you almost always want to sauce it right at the end. If your technique and ingredients are good then the meat, not the sauce, should be your main attraction.
It all depends on the sauce. If it's thick, or if it contains a lot of sugar, then it can scorch over high heat. For something like that, I'd put the sauce on at the very end, or serve it on the side. If it's thinner (something based on wine, vinegar, fruit juice, soy sauce, etc) then it's absolutely OK to put it on before grilling, or marinate the chicken overnight.