The distribution and properties of the first galaxies and quasars are critical pieces of the puzzle in understanding galaxy evolution and cosmic reionization. Previous studies have often excluded unresolved sources as potential low-redshift interlopers. We combine broadband color and photometric redshift analysis with morphological selections to identify a robust sample of candidates consistent with unresolved point sources at redshifts z similar to 8 using deep Hubble Space Telescope images. We also examine G141 grism spectroscopic data to identify and eliminate dwarf star contaminants. From these analyses, we identify three bright (M (UV) less than or similar to -22 AB mag) dropout point sources at 7.5 < z < 8.1. Spectral energy distribution analyses suggest that these sources are either quasars or compact star-forming galaxies. The flux captured by the IRAC 4.5 mu m channel suggests that they have moderate H beta + [O III] equivalent widths. We calculate the number density of point sources at z similar to 7-8, and find that a double-power-law model well describes the point-source distribution. We then extend our analysis to estimate the combined point-source + galaxy luminosity function and find that the point sources have a nonnegligible contribution to the bright-end excess. The fact that the point sources dominate only at M (UV) less than or similar to -22 suggests that their contribution to cosmic reionization is likely limited. While spectroscopic follow-up is needed to confirm the nature of these point sources, this work demonstrates that the inclusion of Lyman-dropout point sources is necessary for a complete census of early galaxies at the epoch of cosmic reionization.