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Two-dimensional (2D) topological insulators (TIs), which can be integrated into the modern silicon industry, are highly desirable for spintronics applications. Here, using first-principles electronic structure calculations, we show that the Au/Si(111)-3 substrate can provide a platform for hosting 2D TIs obtained through the formation of surface alloys with a honeycomb pattern of adsorbed atoms. We systematically examined elements from groups III to VI of the periodic table at 2/3 monolayer coverage on Au/Si(111)-3, and found that In, Tl, Ge, and Sn adsorbates result in topologically nontrivial phases with band gaps varying from 0 to 50 meV. Our scanning tunneling microscopy and low-energy electron diffraction experiments confirm the presence of the honeycomb pattern when Bi atoms are deposited on Au/Si(111)-3, in accord with our theoretical predictions. Our findings pave the way for using surface alloys as a potential route for obtaining viable 2D TI platforms.