Results are presented from data recorded in 2009 by the PHENIX experiment at the Relativistic Heavy Ion Collider for the double-longitudinal spin asymmetry, A(LL), for pi(0) and eta production in root s = 200 GeV polarized p + p collisions. Comparison of the pi(0) results with different theory expectations based on fits of other published data showed a preference for small positive values of gluon polarization, Delta G, in the proton in the probed Bjorken x range. The effect of adding the new 2009 pi(0) data to a recent global analysis of polarized scattering data is also shown, resulting in a best fit Delta G(DSSV)([0.05,0.2]) = 0.06(-0.15)(+0.11) in the range 0.05 < x < 0.2, with the uncertainty at Delta chi(2) = 9 when considering only statistical experimental uncertainties. Shifting the PHENIX data points by their systematic uncertainty leads to a variation of the best-fit value of Delta G(DSSV)([0.05,0.2]) between 0.02 and 0.12, demonstrating the need for full treatment of the experimental systematic uncertainties in future global analyses.