games PCs play (or, hey dawg, I herd u like games; or, "you're all in a bar when" as character generation)

From the Gies' Life In a Medieval City:

Parlor games are played, too, such as those described in Adam de la Halle's Jeu de Robin et de Marion. In 'St.-Cosme' one player represents the saint and the others bring him offerings, which they must present without laughing. Whoever falls victim to his grimaces must pay a forfeit, and become St.-Cosme himself. In another game, 'The King Who Does Not Lie,' a king or queen chosen by lot and crowned with straw asks questions of each.

One gathers that the medievals would have greatly enjoyed a round of Mafia or Werewolf. The medievals, of course, took all the hardcore nerds and threw them into monasteries. The results may well be epitomized by the game of Obligationes:

The most widely discussed kind of obligatio, both in the medieval literature and in recent scholarship, is no doubt positio or “positing.” In a positio the “opponent” begins by saying “I posit that p.” The proposition p is called the “positum.” The “respondent” then says either “I admit it” or “I deny it,” depending on certain conditions. For example, the treatises by Roger Swyneshed, Robert Eland, and Richard Lavenham all stipulate that in order for the positum to be “admissible,” it must be a contingent proposition.^[10] Other authors, as we have seen above, allow the positum to be an impossible proposition, provided that its impossibility is not “manifest,” so that the proposition can be entertained and believed.^[11] We shall not consider such cases here.

If the positio is admitted, the disputation is under way: The opponent then “proposes” to the respondent a series of propositions, one after another. Admitting a positio is not the same thing as conceding a proposition. By admitting a positio the respondent accepts to having the duty to concede the positum if it is put forward as a propositum, and any other duties deriving from this duty. In practice this means that the respondent must evaluate each proposition, or propositum put forward by the opponent with recognition of the duty accepted by admitting the positio. To each propositum put forward by the opponent the respondent must reply by saying “I concede it,” “I deny it,” or “I doubt it,”. The correct response is guided by rules given in the treatises, and different authors give somewhat different rules. Below we discuss the two most important approaches to the rules, called “the old response” and “the new response” by Eland. They can be identified as the rules given by Walter Burley and Roger Swyneshed. ^[12]

According to both of these sets of rules, the correct response depends in part on whether the propositum is “relevant” or “irrelevant” (pertinens/impertinens), and if it is relevant whether it is “sequentially relevant” or “incompatibly relevant” (pertinens sequens/pertinens repugnans). The specification of these notions and of how they affect the correct response to the propositum constitutes the kernel of the theory of positio, and varied from author to author.

Given the variance in the ways different authors give the rules, it may appear that the rules are stipulated, and the disputation appears like a game. On this reading, the participant would have to agree at the beginning which set of rules are followed in the incipient disputation. 

All of this lends great plausibility to the lovely framing conceit of Wolves of God (that is it a recovered manuscript of an RPG played by ye Anglo-Saxons.)

What is the use of this for your own (cod-medieval tabletop roleplaying) play? I can think of a few:

1. To introduce characters to each other and build backstory. It's a D&D cliche, perhaps the D&D cliche, for characters to meet at a bar. If characters haven't been generated collectively, this may lead to an awkward series of introductions. "I'm, uh, Gundnar the Dwarf. I'm very superstitious." Drinking and parlor games were invented to solve precisely this problem! The party could introduce each other through a games of "Two Truths And A Lie" or "Nought Ere Hath I." 
2. To break things up or provide an excuse for a quiet roleplaying moment. Maybe you just want to play a round of Werewolf instead. Okay - have the PCs play the local equivalent of werewolf as they get a daily rest in around the campfire. 
3. To show off. RPGs and parlor games are both exercises in roleplaying. By going one level deeper, you're having fun pretending to be somebody pretending to be somebody, and there are potentials for deep roleplaying with this. Maybe you would deduce that Alice is lying, but Gundnar the Dwarf wouldn't. Maybe your PCs are impersonating nobility and are then invited to a parlor game in which the layers go three levels deep! Go wild!

Examples like the Gies' above show that you don't have to be too creative with the game itself - probably any society that drinks would get a kick out of Never Have I Ever or Truth or Dare. But if you can build out the specifics, then so much the better.