We use first-principles electronic structure calculations to predict a new class of two-dimensional (2D) topological insulators (TIs) in binary compositions of group III elements (B, Al, Ga, In, and Tl) and bismuth (Bi) in a buckled honeycomb structure. We identify band inversions in pristine GaBi, InBi, and TlBi bilayers, with gaps as large as 560 meV, making these materials suitable for room-temperature applications. Furthermore, we demonstrate the possibility of strain engineering in that the topological phase transition in BBi and AlBi could be driven at ∼6.6% strain. The buckled structure allows the formation of two different topological edge states in the zigzag and armchair edges. More importantly, isolated Dirac-cone edge states are predicted for armchair edges with the Dirac point lying in the middle of the 2D bulk gap. A room-temperature bulk band gap and an isolated Dirac cone allow these states to reach the long-sought topological spin-transport regime. Our findings suggest that the buckled honeycomb structure is a versatile platform for hosting nontrivial topological states and spin-polarized Dirac fermions with the flexibility of chemical and mechanical tunability.