Hit dice in the original little brown books only used six sided dice. Plusses were applied to modulate the values by class. So (ignoring constitution for simplicity), a fourth level fighter would have 4d6 HP, whereas a fourth level magic-user would have 2d6+1 HP. Coming to OD&D from one of the later editions, the absence of different hit dice is one of the most striking differences (along with flat d6 weapon damage, to which the size of hit dice is related). This approach was dropped almost immediately, in Supplement I: Greyhawk, and forever after different classes have used different sized dice for their hit points.
I have come to appreciate the hit dice being only from six siders. It keeps the totals grounded. It works particularly well with rerolling per session (the scalar bonuses feel less strange, as they just modify the per-session expected value slightly). However, the actual progression seems somewhat arbitrary and unmotivated. For example, the original fighter “dice for accumulative hits” progression was, by level:
What’s up with that?
It’s relatively easy to fix, though. Here is my proposal, broken down into progressions for high (fighter), medium (cleric), and low (magic-user) hit dice.
The patterns should be easily intelligible. The expected values are relatively close to the originals, though in Men & Magic the magic-user gets all the way up to 9+2 (by 16th level). Gary sure did love the wizards.
What about after tenth level? My inclination would be to halt HD growth entirely past tenth level. A sort of soft E10. It would also be reasonable to keep adding some bonus HP though, in the manner of AD&D. I would probably just make it a flat +1 per level for every level above ten, irrespective of class. Going much above +1 would start to overwhelm the d6 values, unlike the larger hit dice sized used by AD&D, so probably best not to adopt the AD&D numbers (given characters of equal level, the relationship of fighter > cleric > magic-user will still obtain, which I think is the most important thing).