should it always basically be connected to Vcc?
That's the norm. If you're picky about noise you can connect through a coil to provide an LC network.
Using a resistor would drop a small amount of Vs (you'd have to know how much current the ADC draws), I doubt it would matter but I can't see much point.
Maybe you should tell us why you want to fly in the face of convention
Or is there ever a time it would want it's own power (different voltage or noise immunity)?
I've thought about doing that, using a precision regulator or even a voltage reference if the power required is small enough. However you have to be careful that VCC and AVCC are always within 300mV of each other. I'm not sure what happens if that aren't but I suspect it's not good.