Hull defines the conversion factor for a bond as the "quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum."
My understanding is that the hypothetical bond underlying the futures contract has a 6% coupon, and that the (quoted) price of this bond varies with the zero curve. If this zero curve is flat and equal to 6% for all maturities, then the quoted price of the hypothetical bond should be 100 and the product of this hypothetical price and the conversion factor for a particular bond is exactly the quoted price of the bond in question. However, if the zero curve is flat and not equal to 6%, or more generally is not flat at all, then why should this conversion factor be considered a good approximation of the quoted price of the actual bond?
Suppose for the moment that the zero curve is flat. Let P(N,y,c) be the present value of a bond with semiannual coupon c and time to maturity N when the zero rate is y (for all maturities). The present value of the hypothetical bond is P(N,y,0.06). What naturally seems to be the correct conversion factor to get the quoted price of a bond with coupon c and time to maturity M is
CF = P(M,y,c)/P(N,y,0.06).
When y=0.06, this conversion factor is the same one defined by Hull, but otherwise they need not be the same. Is there some reason why it is assumed that the conversion factor is constant? y need not be close to 0.06, and M can be different from N, so it doesn't seem clear that the (constant) conversion factor gives anything useful.