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In the baseline design for advanced LIGO interferometers, the most serious noise source is tiny, dynamically fluctuating bumps and valleys on the faces of the arm-cavity mirrors, caused by random flow of heat in the mirrors' sapphire substrates: so-called "thermoelastic noise". We propose replacing the interferometers' baseline arm-cavity light beams, which have Gaussian-shaped intensity profiles, by beams with mesa-shaped profiles that are flat in their central ~7 cm of radius, and that then fall toward zero as quickly as is allowed by diffraction in LIGO's 4 km arms. The mesa beams average the bumps and valleys much more effectively than the Gaussian beams. As a result, if the beam radii are adjusted so diffraction losses per bounce are about 10 ppm, replacing Gaussian beams by mesa beams reduces the thermoelastic noise power by about a factor 3. If other thermal noises are kept negligible, this will permit advanced LIGO to beat the Standard Quantum Limit by about a factor 1.5 over a bandwidth about equal to frequency, and the event rate for inspiraling neutron star binaries will increase by about a factor 2.5. The desired mesa beams can be produced from input, Gaussian-profile laser light, by changing the shapes of the arm cavities' mirror faces from their baseline spherical shapes (with radii of curvature of order 60 km) to Mexican-hat (MH) shapes that have a shallow bump in the center but are otherwise much flatter in the central 10 cm than the spherical mirrors, and then flare upward strongly in the outer 6 cm, like a sombrero. We describe mesa beams and MH mirrors mathematically and we report the results of extensive modeling calculations, which show that mesa-beam interferometers are not much more sensitive than the Gaussian-beam interferometers to errors in mirror figures, positions, and orientations.