If Hirelings are your basic employees, henchmen are loyal (to a degree) followers of the character. They could be thought of as sidekicks, apprentices, friends, etc. The DMG claims that henchmen "are greatly desired by discerning players, for they usually spell the difference between failure and success in the long term view" (DMG 34). An interesting statement, given our earlier discussion of how the outlook of players and play style has changed over the years; I can't remember ever having a PC with a henchman, and I've been playing since 1979...though back in junior high we did (as did, I suspect, a lot of people) go through a phase where everyone ran 2 or 3 characters at a time.
This section breaks down everything you ever wanted to know about henchmen...and a lot of stuff you probably just didn't need to know. Like the hirelings section, of which it is a logical extension, this section is likely one of the best, most detailed, and most useful in the book, and one that has sadly been lost in future editions.
Level of Henchmen: It begins with a section discussing that, as a general rule, henchmen will be first level characters, though as the main PCs rise in experience, there is a flat-percentage chance that they can draw higher level henchmen, up to third level of experience.
Role Playing and Flavor: Following this, we get several sections on the race of henchmen; from locale and racial distribution to racial specifications, which basically says that if you're in elven lands, you'll be more likely to draw an elf follower, etc.
Next we get sections on the numbers, effective locations (how to find them), length of time required to "screen" applicants, and etiquette for treatment of prospective henchmen during the process. We then get a breakdown of the distribution of character classes that will apply. Far more than just demographic lists, there's information here on how to post ads, proper etiquette and treatment of applicants, time and procedures that applicants might follow, and a lot of actual "game world" discussion.
There's actually a great deal of flavor in the advice here that is uncommon to the DMG; it seems clear (if one is familiar with the origins) that these sections are being written from the perspective of Gygax's Greyhawk game. This isn't a bad thing, to my mind, as Greyhawk did become the standard by which most first edition games were measured. "Generic AD&D" and "Greyhawk" became non-implicit synonyms by the early-mid-80's. That is to say, the books never made it explicit that if you were playing AD&D without a campaign setting, you were playing in Greyhawk, but exchanges similar to the following were often heard:
"What campaign setting are you using?"
"None. Just generic."
"Ah, so Greyhawk, then."
"What campaign setting are you using?"
"I guess you could say Greyhawk. But I don't own any of the stuff. Just kind of making it up as we go."
Later, in third edition, we'd see this become explicit: the "default" was Greyhawk.
Cost of Successful Employment: Though more loyal and devoted, and often more fleshed-out than hirelings, Henchmen are employees and do require "upkeep." To use a chess analogy, if the PC's are the king and queen, then hirelings are pawns, while henchmen are rooks, knights, and bishops. Hirelings carry your gear and fix your armor; henchmen are the guys and gals who get your back and expect (and deserve) better treatment and pay as a result. This section really breaks down the costs associated with henchmen, covering initial salary, equipment, quarters and support, and activity and shares to which they are entitled. Offering more or less will increase or decrease the percentage chance that the prospective henchman will accept employment.
Finally, all of the above is further modified somewhat by the henchman's characteristics, for which the DM is pointed to the Personae of Non-Player Characters section, which oddly is not found for another 65 pages. It strikes me that this section should have been included directly following the sections on hirelings and henchmen, as this is where such information would've been most often and immediately useful, particularly since the occupational information included in these two sections is essential for the creation of general NPCs as well as those in the employ of the PCs.
When all is said and done, the PC may decide to make an offer, after which (assuming the DM hasn't already planned this as part of the game) percentile dice are rolled against the prospect's final interest level to see if the offer is accepted.
The more I read, the more it becomes clear that standard combat notwithstanding (we'll deal with unarmed combat later), AD&D used a straight percentile roll-under system to resolve almost everything, and it's actually fairly elegant the way its handled, if not exactly the most intuitive (i.e. it does require consulting modifier charts for determining chance of success).
The elegance comes from the fact that the chance of success takes all opposing elements into account and comes up with a chance to succeed that doesn't result in defensive capabilities becoming uberstats, as is all-too-often the case in games such as BRP, whereby if you have a high dodge score, no one will ever hit you, because dodge always trumps attack.
Agree or disagree as you will: that's my story, and I'm stickin' to it!
Oh, also, earlier someone mentioned ability checks. As of this point in the game, ability checks did not yet exist as part of the system. There are no "roll under your ability" rules in the PHB or DMG. That came later, with the introduction of Nonweapon Proficiencies in Oriental Adventures, and we'll cover it when it arises.
Wow, I'm doing a lot of commentary in this section.
Anyway, the section on Henchmen ends with a bit on Loyalty of Henchmen & Hirelings, Obedience and Morale. Here we get back to the lists of tables and charts from earlier, breaking down base morale levels, situations in which morale and obedience checks are required, modifiers to morale, obedience and loyalty, etc. Standard fare, but undeniably useful and good stuff. Once again, at the end of this section, we get a cross-reference--this time to the Combat section (which appears much later).