  The increasing neutrality of the intergalactic medium at z>6 suppresses
Ly-alpha emission, and spectroscopic confirmation of galaxy redshifts requires
detecting alternative UV lines. The strong [C III] 1907 + C III] 1909 doublet
frequently observed in low-metallicity, actively star-forming galaxies is a
promising emission feature. We present CLOUDY photoionization model predictions
for C III] equivalent widths (EWs) and line ratios as a function of starburst
age, metallicity, and ionization parameter. Our models include a range of C/O
abundances, dust content, and gas density. We also examine the effects of
varying the nebular geometry and optical depth. Only the stellar models that
incorporate binary interaction effects reproduce the highest observed C III]
EWs. The spectral energy distributions from the binary stellar population
models also generate observable C III] over a longer timescale relative to
single-star models. We show that diagnostics using C III] and nebular He II
1640 can separate star-forming regions from shock-ionized gas. We also find
that density-bounded systems should exhibit weaker C III] EWs at a given
ionization parameter, and C III] EWs could therefore select candidate Lyman
continuum-leaking systems. In almost all models, C III] is the next strongest
line at < 2700 Angstroms after Ly-alpha, and C III] reaches detectable levels
for a wide range of conditions at low metallicity. C III] may therefore serve
as an important diagnostic for characterizing galaxies at z>6.
