One thing about learning many games (here, I am talking about adventure games/RPGs, or "tabletop" RPGs, since now we have to accept that there's a type of computer game that adopted the name but has gained currency over the original usage of the term) is that you'll almost always find some small rule in a game that makes sense and seems fun, but isn't included in most other games. A lot of these are famous, such as the Personality Trait rules in Pendragon or the Sanity rules in Call of Cthulhu, but others are less well-known. I'll do a quick overview of a few of these.
In the Traveller supplementary game, Snapshot, there is a list of actions that a character can take each game turn by spending Action Points. Turns are about 15 seconds long and an average character has a range of from 2 to 30 APs for human characters with no enhancements, and an average of 14 APs. That is roughly 1-2 AP per second or a bit less, in the game turn (but that's not relevant here). During a turn, a character can try to pick up an object that has been dropped. To do so, it takes a random number of AP, generated by rolling a d6. If the player wants, the character can abort this by spending 1 AP, but may not attempt to pick up the object again until a different action is taken. One of the actions permitted is the 1 AP action called "Expletive", which is pretty descriptive. So, a character might reach for a dropped knife, but not get a good grip, draw back and swear, then attempt to pick it up again. Alternate uses for "Expletive" include filling time while a machine, such as an automatic door for example, cycles through its action. So, a door might take 3 APs to open, requiring the character to spend 3 AP before going through the door. Sure, some other action might be taken, but if nothing else presents itself, the character can spend three consecutive "Expletive" actions, costing a total of 3 AP.
Sticking with Traveller-related games, let's discuss the Jack-of-All-Trades skill. In most versions of the game, JoT (the usual abbreviation) alleviates the penalty for not having a skill. That is, if a skill is not possessed by the character, usually they will take a penalty to attempts to perform actions using that skill, varying slightly by particular edition. With JoT, that penalty is reduced or alleviated in some fashion (for example, in Marc Miller's Traveller, or T4 for short, if the difficulty of the action is greater than the character's skill level plus their JoT level, then the difficulty is increased by one; in others, the numeric die roll penalty of -3 or -4 for lack of skill is reduced by the JoT level toward 0, but never providing a bonus). To my way of thinking, that does a poor job of representing what JoT skill actually represents in the setting. MegaTraveller (MT is the official abbreviation) took a very different tack with the JoT skill. Rather than providing a situational bonus to characters who lack a skill, it tried to simulate flexible thinking. In MT, there is an economy of re-attempting a failed skill roll, assuming that it isn't an urgent skill like an attack or attempt at social influence. If a character normally fails an attempt, which is a failure by 1, they can go ahead and retry, but of course this takes as much time as the original attempt. However, if they have an "Exceptional" failure (missing the roll on 2d6 by 2 or more), they normally must roll to "Stay Determined", that is to say they need to be able to come up with a new approach to the problem. This roll is based on the character's Intelligence and Education attributes. The advantage given by JoT skill is that they get a number of free attempts to retry a task equal to the JoT skill level, representing greater resourcefulness at approaching the task. Note that this means that JoT is not typically useful for the types of urgent skill uses described earlier, but that's kind of the point. The skill objectively improves the ability of any character, skilled or not, to perform technical, research, crafting, and similar actions by giving them more attempts to succeed, assuming they have the time to spend. This represents coming up with different approaches to the problem. It doesn't do this by simply increasing the character's effective skill level (even if fractionally) as the other editions do.
Moving on to GURPS 4th edition, there is an optional rule called "The Last Gasp", found in Pyramid magazine issue 3/44: Alternate GURPS II. This actually comprises two separate sections that revise the Fatigue rules in GURPS, but I'm only going to deal with the second section, "Short-Term Fatigue" (there's also a lot to love about the first section, "Long-Term Fatigue", though). In this optional rule, characters get a number of Action Points at the start of combat, based mainly on their attributes. In addition, they get a small pool of points for Skills they have at a high level that can only be used for actions using that Skill, but this pool can only be recovered when the normal AP pool is fully recovered. In this system, unlike the Snapshot APs above (or the APs in FASA's Star Trek and Doctor Who RPGs, which are very similar to the Snapshot ones), APs have to be actively recovered by taking what amount to resting actions during combat, mainly the Do Nothing, Evaluate, and Wait actions (though with a Wait, APs can only be recovered if the Wait is not triggered during the turn). An important exception is that the player may choose to expend regular Fatigue Points (FPs) in exchange for a number of APs recovered (and I think that this should replace the regular system of spending 1 FP for any fight that lasts 10 or more seconds/turns). The system isn't fully developed as printed, with a notable issue being how movement and AP interact, but serves as an excellent basic system for modeling certain types of emergent behaviors that occur during real fights.
In Lace & Steel, one of the skills a character may have is Travel. This skill is used to avoid minor mishaps on the road, ranging from fatigue, embarrassment, lost temper, and the like to rips in clothing (that must be repaired with the Tailor skill or buying new clothing), lost items ("What in the world did I do with that bottle of ink?"), minor injuries like sprains or small animal bites, and so on. Each of these mishaps is given a specific penalty (becoming "disheveled" might be critical if one, say, meets the Duke at a wayside inn on the way to the royal court due to the deleterious effects on Charisma) and a specific way to alleviate it. This also provided reasons to stop at a comfortable waystation like an inn when possible, and to pay more for more comfort, rather than sleeping outside all of the time.
Fantasy Wargaming has an extensive and interrelated magic and religion system that has many interesting features, but I'm only going to deal with one small one here. In the Norse religion section, deities are described in part by their relationships with other deities (this one is the son of so-and-so, this one is married to that one, another is close friends or bitter enemies of yet another, and so on). This affects the ability of a deity to intercede with another on behalf of a character. Normally, of course, one would simply approach the deity in question, but perhaps they have a "Resistance to Appeals" characteristic that is too high, or there is a specific penalty related to that character and that deity for some reason. In such cases, a character might want instead to appeal to another, more sympathetic deity (or perhaps even one's own personal fylgja, which is something not entirely unlike a Guardian Angel, though perhaps not quite the same either) to ask the original deity for a favor. This allows for much more personal nuance to affect the religious actions of characters in the game and helps bring out the personalities of the deities in ways that less flexible, more mechanical systems do not. To an extent, RuneQuest dealt with similar issues by listing "Allied Deities" and the spells that they provide to followers of the main deity, but in a way that was much more rigid than the FW approach allowed.
Do you have any favorite small rules? What games are they found in?