We introduce the concept of Floquet odd-frequency superconducting pairs and establish their emergence in time-periodic conventional superconductors. We show that these exotic Cooper pairs are possible because the Floquet modes in time-periodic systems provide an additional index (a Floquet index) that broadens the classification of superconducting pair symmetries, with no analog in the static regime. Our results thus put forward a different route for odd-frequency superconducting pairs, and pave the way for Floquet engineered dynamical superconducting states.