How can I animate a part of the my mesh to act almost like stiff hair?
Generally, physics. Soft body physics with high goal weight. Use a different, non-rendering mesh and use something like a mesh deform to have it move your (presumably higher poly) rendering mesh.
The Jiggle Bones / Jiggle Armature addon might help here.
Hmm… that might be a bit tricky. Is there a good tutorial that an inexperienced user as myself could follow?