OD&D isn't my platonic ideal for D&D; I started with the 1981 Red Box, and the Basic and Expert D&D books are still the first love. But if there's one global rule I'd consider applying to BX or a retro clone, it'd be to get rid of variable hit dice and use nothing but 6 sided dice - the glorious d6.
I have bags and bags of d6's, tons of them from years playing Axis & Allies and similar games. Consider the simplicity an all d6 game provides for encounters and combat. 10 orcs? No problem, roll 10 D6's at the start of combat - there are your hit points. The same 10 dice become your 10 HD monster - you track hit points by removing dice or adjust values as the creature takes damage. Always rolling hit points at the table and tracking them right on the dice is immensely appealing; I'd never write down hit points again.
It seems easy enough to put some class variability into player HD using d6's; perhaps it's as easy as fighters get a +1, magic users get a -1. The only thing holding me back for now is unwillingness to get rid of variable weapon damage. The extreme example your hear about 'from the old days' is a group arming themselves with iron spikes and rocks (everything does a d6!) but there's also a concern with 2-handed weapons; I've seen house rules like exploding 6's for 2-handed weapons, or double dice for 2-handed weapons, or a simple +1 to damage for 2-handers.
I haven't talked to my players about this secret dream of an all d6 D&D game. The allure of handfuls of square dice runs directly contra to the baroque mystique of using the funny sided polyhedrals.