Why is benzyl alcohol a slightly stronger acid than cyclohexanol?
(This CW answer is adapted from an earlier comment by ron.)

In both cases, the carbon directly attached to oxygen is $\mathrm{sp^3}$ hybridised. However, the next carbon in the two compounds have different hybridisations. In cyclohexanol, it is also $\mathrm{sp^3}$ hybridised; but in benzyl alcohol, it is $\mathrm{sp^2}$ hybridised.

In general, $\mathrm{sp^2}$ hybridised carbons are more electronegative than $\mathrm{sp^3}$ hybridized carbons. This is because the electron density in an $\mathrm{sp^2}$ orbital has more s-character (50%) compared to one in an $\mathrm{sp^3}$ orbital (33%), and is consequently more strongly attracted to the carbon nucleus. Therefore, the carbon attached to oxygen in benzyl alcohol will be more electron deficient (more positive) and will be better able to stabilise the negative charge on oxygen in the alkoxide conjugate base, via the inductive effect.