They absolutely don't.
It's the weapon that does the damage, not the mech.
This can be double-checked in the following way.
It takes 2x Alphas to Headshot a mech with a 30-DMG alpha.
Check this with a DC'ed Jenner or Commando, using both a Light and a Medium.
You won' see any difference between the two.
I've done this multipile times with different mechs (not because of my tin-foil hat, but to spare a mate from being pug-***** when he DCs/Blackscreens/etc).
The person who made that comment was just trying to be a smartarse, and you took it seriously.
*EDIT* The reason the guy above is seeing it take more hits to down a light is due to netcode/hitox de-sync.
I've had AC20 shots hit a Light and do no damage (as they've hit other mechs and do none).
Also, you can occasionally hit a mech, see the red-reticle hit confirmation, but still do no damage, as indicated by the paper doll.
This is true across all mechs, but because everyone loves to think Lights are immune to pain, these instances stick in the mind vs Lights, whereas they get forgotten against Mediums & Heavies.
Edited by BigJim, 03 December 2012 - 06:57 AM.