We have measured the beam-normal single-spin asymmetry ${A}_{n}$ in the elastic scattering of 1--3 GeV transversely polarized electrons from $^{1}\mathrm{H}$ and for the first time from $^{4}\mathrm{He}$, $^{12}\mathrm{C}$, and $^{208}\mathrm{Pb}$. For $^{1}\mathrm{H}$, $^{4}\mathrm{He}$, and $^{12}...