Adding Classes: To add a new class, a character needs an ability of 16 or more in the new prime requisite, and must sacrifice their current top class level. Elves automatically start with two classes.
Experience: At the start of each adventure, the character specifies which class they are training in, and all XP is awarded to that class. Normal restrictions apply (at most one level per adventure).
Benefits: The multiclassed character uses the best entry for hit dice, attacks, and saves; they can freely use any other abilities (weapons, armor, skills, spells, etc.). Fighter/Wizards may cast spells in leather or chain mail, but not plate; thief skills are restricted to leather armor only.
Monsters: Monsters may use the same rules, treating base hit dice as Fighter levels with 6-sides.
A few comments follow. First, keep in mind that I now use Greyhawk-style Thieves to replace Clerics (albeit with d6 hit dice and skills simply rolled d20 + level + Dex >= 20).
I felt deeply that when adding a class, something had to be sacrificed in exchange for this benefit, and this price had to be fairly precious at any level (hence no fixed XP cost). The result was the "sacrifice their current top class level" clause, which I think is fair. In some sense this is similar in spirit to 3E multiclassing, but much less of a full-system overhaul.
The "specify which class they are training in" at adventure-start for all XP was an attempt to retain the spirit of the "freely switch class... but not during the course of a single game" from the original Elf description of multiclassing.
Using only the highest hit die type may seem odd in retrospect, but I've written before as to how it's elegantly consistent with all the other "max" operators for multiclass abilities in OD&D/AD&D. The implication is that you should keep separate hit point scores for each class, and only use whichever one is currently highest.
Finally, here's the reason for allowing Fighter/Wizards to use leather/chain but not plate: While the OD&D Elf description allows them to "use magic armor", for some mysterious reason, Elves never appear in plate in the original rules. The Vol. 2 monster entry has them all in chainmail. The Swords & Spells entry has the heaviest-armored Elf classification wearing chainmail. (And likewise 1E Bards can cast in chainmail but not heavier types.) So let's say that chain is broadly conducive to spellcasting, but plate is prohibited. It seems both consistent with the source material, and a useful and elegant game-balance restriction.