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LINER nuclei, which are present in many nearby galactic bulges, may be the manifestation of low-rate or low-radiative-efficiency accretion onto supermassive central black holes. However, it has been unclear whether the compact UV nuclear sources present in many LINERs are clusters of massive stars, rather than being directly related to the accretion process. We have used the Hubble Space Telescope to monitor the UV variability of a sample of 17 galaxies with LINER nuclei and compact nuclear UV sources. Fifteen of the 17 galaxies were observed more than once, with two to five epochs per galaxy, spanning up to a year. We detect significant variability in most of the sample, with peak-to-peak amplitudes from a few percent to 50%. In most cases, correlated variations are seen in two independent bands (F250W and F330W). Comparison to previous UV measurements indicates, for many objects, long-term variations by factors of a few over decade timescales. Variability is detected in LINERs with and without detected compact radio cores, in LINERs that have broad Halpha wings detected in their optical spectra (``LINER 1s''), and in those that do not (``LINER 2s''). This variability demonstrates the existence of a nonstellar component in the UV continuum of all types and sets a lower limit to the luminosity of this component. Interestingly, all the LINERs that have detected radio cores have variable UV nuclei, as one would expect from bona fide active galactic nuclei. We note a trend in the UV color (F250W/F330W) with spectral type-LINER 1s tend to be bluer than LINER 2s. This trend may indicate a link between the shape of the nonstellar continuum and the presence or the visibility of a broad-line region. In one target, the poststarburst galaxy NGC 4736, we detect variability in a previously noted UV source that is offset by 2.5" (~60 pc in projection) from the nucleus. This may be the nearest example of a binary active nucleus and of the process leading to black hole merging. Based on observations with the Hubble Space Telescope, which is operated by AURA, Inc., under NASA contract NAS 5-26555.