Measurements of the linearly-polarized photon beam asymmetry Σ for photoproduction from the proton of η and η′ mesons are reported. A linearly-polarized tagged photon beam produced by coherent bremsstrahlung was incident on a cryogenic hydrogen target within the CEBAF Large Acceptance Spectrometer. Results are presented for the γp→ηp reaction for incident photon energies from 1.070 to 1.876 GeV, and from 1.516 to 1.836 GeV for the γp→η′p reaction. For γp→ηp, the data reported here considerably extend the range of measurements to higher energies, and are consistent with the few previously published measurements for this observable near threshold. For γp→η′p, the results obtained are consistent with the few previously published measurements for this observable near threshold, but also greatly expand the incident photon energy coverage for that reaction. Initial analysis of the data reported here with the Bonn-Gatchina model strengthens the evidence for four nucleon resonances — the N(1895)1/2−, N(1900)3/2+, N(2100)1/2+ and N(2120)3/2− resonances — which presently lack the “four-star” status in the current Particle Data Group compilation, providing examples of how these new measurements help refine models of the photoproduction process.