Morale is one of the things that's very unclear in OD&D. It's discussed briefly on page 13 of the Men & Magic booklet, where it says "Non-player characters and men-at-arms will have to make morale checks (using the above reaction table or "Chainmail") whenever a highly dangerous or un-nerving situation arises." Unfortunately, there isn't an NPC reaction table in the book, so by default Chainmail would have to be the place to look. Unfortunately morale in Chainmail is an arcane algebra that makes my eyes glaze over.
Luckily, a few weeks ago Christian Lindke at the Cinerati blog made a really great post about the history of morale in D&D. In the post he dissects morale in Chainmail to identify two parallel morale systems: Melee Morale and Casualty Morale. Melee Morale is a very complex set of rules for large-scale combat that would be useless in D&D, but it turns out that Casualty Morale is very similar to the morale rules given in Basic/Expert D&D c.1981. It's very likely that the Chainmail Casualty Morale rules are what's being referred to in OD&D, and in fact they probably served as the precursor for morale rules in Basic/Expert D&D. In a nutshell, NPCs have a morale value between 2-12 where high values mean better morale. In stressful situations the DM rolls 2d6. If the roll is higher than the morale score, the NPC freaks out and flees.
Okay - easy. So how do we apply this to henchmen in OD&D? Luckily, the rules are a little more specific on this...
"When one or more such characters are taken into service a loyalty check is made by rolling three-six-sided dice. Adjustments are made for charisma and initial payments for service, and the loyalty of the character(s) noted by the referee. (The player will not have any knowledge of what it is without some method of reading minds.)"
Loyalty Score Morale
3 or less Will desert at first opportunity
4-6 -2 on morale dice
7-8 -1 on morale dice
9-12 Average morale dice
13-14 +1 on morale dice
15-18 +2 on morale dice
19 and above Need never check morale
So this is what I will do...
1. For each prospective henchman roll 2d6 to generate a secret morale score.
2. When a PC hires a henchman roll 3d6 on the Loyalty table.
3. Adjust the Loyalty Score based on the PC's Charisma modifier and additional monetary incentives.
4. Modify the henchman's morale according to the Loyalty Score.
5. Make secret 2d6 morale checks for each henchman during stressful situation (e.g. first casualty, first wounding, etc.)
I'm sure someone else has worked through this before on some forum or blog somewhere, but I had fun spending a few minutes thinking about it. Filling in the blanks is part of the fun of OD&D. I'll let you know how it goes!