We present a comprehensive survey of CII* absorption detections toward stars within 100 pc in order to measure the distribution of electron densities present in the local interstellar medium (LISM). Using high spectral resolution observations of nearby stars obtained by GHRS and STIS onboard the Hubble Space Telescope, we identify 13 sight lines with 23 individual CII* absorption components, which provide electron density measurements, the vast majority of which are new. We employ several strategies to determine more accurate CII column densities from the saturated CII resonance line, including, constraints of the line width from the optically thin CII* line, constraints from independent temperature measurements of the LISM gas based on line widths of other ions, and third, using measured SII column densities as a proxy for CII column densities. The sample of electron densities appears consistent with a log-normal distribution and an unweighted mean value of n_e(CII_SII) = 0.11^+0.10_-0.05 cm^-3. Seven individual sight lines probe the Local Interstellar Cloud (LIC), and all present a similar value for the electron density, with a weighted mean of n_e(LIC) = 0.12 +/- 0.04 cm^-3. The Hyades Cloud, a decelerated cloud at the leading edge of the platoon of LISM clouds, has a significantly higher electron density than the LIC. Observed toward G191-B2B, the high electron density may be caused by the lack of shielding from such a strong radiation source. Given some simple assumptions, the range of observed electron densities translates into a range of thermal pressures, P/k = 3300^+5500_-1900 K cm^-3. This work greatly expands the number of electron density measurements and provides important constraints on the ionization, abundance, and evolutionary models of the local interstellar medium. (abridged)
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