Odd-frequency (odd-omega) electron pair correlations naturally appear at the interface between BCS superconductors and other materials. The detection of odd-omega pairs, which are necessarily nonlocal in time, is still an open problem. The main reason is that they do not contribute to static measurements described by time-local correlation functions. Therefore, dynamical measurements, which depend on nonlocal time correlations, are suitable for detecting these pairs. In this work, we study the signatures of odd-omega pairs in the supercurrent noise through a weak link between two superconductors at different superconducting phases. We show that the finite-frequency current noise can be decomposed into three different contributions coming from even-frequency (even-omega), odd-omega pair amplitudes, and electron-hole correlation functions. Odd-omega pairing, which is interlead (between electrons at different sides of the junction), provides a positive contribution to the noise, becoming maximal at a superconducting phase difference of pi. In contrast, intralead even-omega pair amplitude tends to reduce the noise, except for a region close to pi controlled by the transmission of the junction.