I revised the default deer to better represent, at least in my opinion, the deer that I am familiar with (namely, the North American white-tail deer).

I made them a little stronger than the ones in the SRD, reduced their bite damage, gave them kick and ram (headbutt) attacks, reduced their speed from 50 to 35 but added the ability to Dash as a bonus action, gave them a keen sense of smell, and gave them an extra hit point. 

For the males, I gave them 1 more point of Strength than the females, an extra hit die, and during summer and autumn (when they have grown antlers), a gore attack.