We predict the transverse momentum (pT) dependence of elliptic flow of thermal photons for Au+Au collisions at the BNL Relativistic Heavy Ion Collider. We model the system hydrodynamically, with a thermalized quark-gluon plasma at early times followed by hadronization and decoupling. Photons are emitted throughout the expansion history. Contrary to hadron elliptic flow, which increases monotonically with pT, the elliptic flow v2(pT) of thermal photons is predicted to first rise and then fall again. Photon elliptic flow at high pT reflects the quark momentum anisotropy at early times when it is small, while at low pT it mirrors the large pion momentum anisotropy during the late hadronic emission stage. An interesting structure is predicted at intermediate pT∼0.4GeV/c, where photon elliptic flow reflects the momenta and the (compared to pions) reduced v2 of heavy vector mesons in the late hadronic phase.