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. Our results focus attention on the necessity of improving the measurement of heavy-element abundances, while at the same time reducing the estimated uncertainties in the predicted solar neutrino fluxes due to abundance errors. 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 the preferred "conservative" (based on changes of measured values with time) and "optimistic" (current values) 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.