  We show that uncertainties in the values of the surface heavy element
abundances of the Sun are the largest source of the theoretical uncertainty in
calculating the p-p, pep, 8B, 13N, 15O, and 17F solar neutrino fluxes. We
evaluate for the first time the sensitivity (partial derivative) of each solar
neutrino flux with respect to the surface abundance of each element. We then
calculate the uncertainties in each neutrino flux using `conservative
(preferred)' and `optimistic' estimates for the uncertainties in the element
abundances. The total conservative (optimistic) composition uncertainty in the
predicted 8B neutrino flux is 11.6% (5.0%) when sensitivities to individual
element abundances are used. The traditional method that lumps all abundances
into a single quantity (total heavy element to hydrogen ratio, Z/X) yields a
larger uncertainty, 20%. The uncertainties in the carbon, oxygen, neon,
silicon, sulphur, and iron abundances all make significant contributions to the
uncertainties in calculating solar neutrino fluxes; the uncertainties of
different elements are most important for different neutrino fluxes. The
uncertainty in the iron abundance is the largest source of the estimated
composition uncertainties of the important 7Be and 8B solar neutrinos. Carbon
is the largest contributor to the uncertainty in the calculation of the p-p,
13N, and 15O neutrino fluxes. However, for all neutrino fluxes, several
elements contribute comparable amounts to the total composition uncertainty.
