The quantum spin Hall (QSH) insulators possess potential applications in dissipationless spintronics. With the rapid development of flexible electronics, it is highly desirable to search or design high-performance QSH insulators with excellent flexibility and large bandgap, which is still a huge challenge. Here, using first-principles calculations, we report that surface oxygen functionalization can transform two-dimensional (2D) GaSe monolayer from a trivial insulator to a nontrivial topological insulator with superior stretchability. The fully O-functionalized GaSe monolayer, GaSeO, exhibits a sizable QSH gap of 178 meV, in which the topological nature is characterized by the nontrivial Z2 topological invariant and the topological edge states. The nontrivial band topology originates from the s-p band inversion in the crystal field. Remarkably, the GaSeO monolayer shows isotropic mechanical flexibility and can sustain large strains of up to 26%, beyond graphene and 2D flexible Ti2C. Furthermore, the QSH state of GaSeO is robust against small elastic strains. The discovery of this 2D flexible and large-gap QSH insulator opens up new opportunities for developing stretchable and low-power electronic devices.