|We investigate the onset of X-ray jets observed by Hinode/XRT. Each jet was near the limb in a polar coronal hole, and showed a ``bright point'' in an edge of the base of the jet, as is typical for previously-observed X-ray jets. We examined SDO/AIA EUV images of each of the jets over multiple AIA channels, including 304 Ang, which detects chromospheric emissions, and 171, 193, and 211 Ang, which detect cooler-coronal emissions. We find the jets to result from eruptions of miniature (size <~10 arcsec) filaments from the bases of the jets. In many cases, much of the erupting-filament material forms a chromospheric-temperature jet. In the cool-coronal channels, often the filament appears in absorption and the hotter EUV component of the jet appears in emission. The jet bright point forms at the location from which the miniature filament erupts, analogous to the formation of a standard solar flare arcade via flare (''internal'') reconnection in the wake of the eruption of a typical larger-scale chromospheric filament. The spire of the jet forms on open field lines that presumably have undergone interchange (''external'') reconnection with the erupting field that envelops and carries the miniature filament. This view is consistent with a number of other earlier high-resolution studies of coronal jets. It differs however from the basic version of the ''emerging-flux'' scenario for X-ray jets. This work was supported by funding from NASA/LWS, Hinode, and ISSI.