People have thrown around many theories about which exact aspects of a person determine their ability to channel. It's hard to pinpoint because the Wheel can often be blamed for "adjustments" of certain parameters to fit with the criteria.

The best example of this is the strength of channelers we know have been re-spun throughout time. The Dragon's soul, for example, seems to carry with it the ability to channel strongly--in two Ages in a row, he's the strongest channeler of the time, across from Ishamael/Moridin (although, in that case, the soul has spanned 2 ages). The Wheel, however, might have specifically selected a body for the Dragon's soul with strong channeling ability--this is why it's hard to say the soul carries any channeling weight. There's some evidence that points to this fact, namely the transmigrated Forsaken. One could easily say that the DO simply found bodies with comparable channeling ability, but the fact that the Forsaken compose some of the strongest channelers in the world suggest that the soul does carry some weight--there might just not be bodies lying around with such ability! Other evidence is that of Aran'gar, whose soul is obviously connected directly to saidin and whose body is thus irrespective of which power i uses.

There's plenty of evidence that there are physical structures in the body that can tune channeling ability. The best, of course, is Siuan and Leanne, who have identical souls as before but differing channeling abilities. Other evidence is Cyndane (why does Brandon pretend we don't know? We saw concrete evidence from Cyndane's PoV duing the Cleansing that Cyndane is Lanfear), who has the same soul, different body, and different channeling ability.

I'm under the impression that the body is responsible for the majority of channeling ability, and the soul is indirectly responsible in that it affects the Wheel's choice of body.