Why is benzyl alcohol a slightly stronger acid than cyclohexanol? Cyclohexanol has a pKa = 16, while benzyl alcohol has a pKa = 15.4 

The given explanation is that the "polarity of the benzyl group stabilizes the negative charge on oxygen". 

My understanding is that when resonance is not a concern, benzyl groups are primarily electron donating by the inductive effect... which would destabilize charge on oxygen. 

I know the inside of the aromatic ring is more negative, and the outside is more positive. Is it this positive charge that stabilizes the negative charge on oxygen?
(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.