I enjoy telling people why I love this game but I keep getting asked "How fiddily is it?" Has anyone else been asked this question? What has been your response? I just try and explain the game the best I can and hope that answers their question.
On a minor tangent, I hate the word "fiddily" as everyone seems to have a different idea of what it means.
For me a game is fiddily when, even when you are familiar with the game, you still make errors on a regular basis. With Sentinels I'm guessing more than half the time I miss some interaction during the course of the game. Even though I understand the rules and have played over 100 times. There is just a lot to keep track of, and it lends to mistakes. (Especially in a Villians game or against Kaargra). At least that is my interpretation of fiddilty.
I don't have Spirit Island yet so I'm not sure how fiddily it is but I'm excited to play it.
One thing I would definitely say is that Spirit Island is a medium to heavy game. Eric told me at Origins that he was aiming for a cooperative game that is just a little bit lighter than Mage Knight. This is a choice, filling an underserved niche (of heavier cooperative games), and is something people should know before playing or buying the game.
I never quite know what is meant by fiddlyness, but maybe let's compare it to Sentinels in terms of keeping track of modifiers, etc. Like Sentinels, there are lots of different cards, each with unique effects. This is mitigated a bit by the fact that in a given game you keep recycling the same cards through your hands.h
It's quite difficult to figure out exactly which powers to use and where you plan to use them.
Once you've made the decision on which powers to use, I find it a bit easier to figure out what happens. The big difference to me is the spatial aspect of Spirit Island and the lack of global effects: Powers may do something funky, but they'll only do it in one land, so there aren't multiple competing things to resolve. The reminder tokens are also very useful for keeping track of things like "where did I use Year of Perfect Stillness, again?"
Of course it is a "figures on a map" game, and there's a certain amount of physical fiddliness associated with that, particularly since the number of figures in a region can get fairly large. But I don't think that's what people are usually complaining about?
I think it's also worth referring people to Eric's BGG post, Why you'll love (or not) Spirit Island. It goes into a lot of detail on things that may be positive or negative for prospective players.
I like what cnranger said. I would also like to add that while not the definition of fiddlyness, fiddly games tend to take away from the amount of time I am actually "playing" the game. Constantly moving tokens, checking for triggers, doing automated things that don't involve my decisions aren't fun either. This also includes trying to find all the information I need to make my decision.
Thankfully, the fiddlyness can be mitigated by A) playing straightforward decks, and B) having one person who really knows the game keeping track of the board. But yeah, it can be somewhat fiddly, depending on who's in play.
Are you saying this about SotM or Spirit Island? This sounds like you talking about SotM
The automated bits in Spirit Island are all in the Invader Phase, which is generally quite sure (although painful). The Event Cards in the expansion do introduce some more triggers and such.
The word feels pretty vague to me; I don't hear or use it often. I like cnranger's description.
The thing in Sentinels that I think leads to that kind of fiddliness is that effects on one card interact with another card. When you are dealing 2 damage, you have to remember all of the damage boosts, reductions, redirections, type modifiers, etc. That often requires looking in play areas other than the dealer and recipient of the damage and doesn't always have a physical representation at all (e.g. Exploit Vulnerability or The Seeker).
With very occasional exception, the powers in Spirit Island are self-contained; when you activate a power, the only relevant thing is the text of that card. Heart of the Wildfire has a power that boosts a spirit's damage. There are a couple that let you increase the range of a spirit's powers for a turn and a couple that let you make slow powers fast or repeat a power. These enduring effects are rare - most effects are either instant or can be represented by one of the reminder/defense tokens. The fear and event cards can introduce enduring effects, but these are also not too common and usually apply to a specific part of the invader phase, so you can just put the card on top of that invader action as a reminder (e.g. put a fear card that grants defense on top of the ravage card).
I still make mistakes from time to time, but far less frequently than in Sentinels.
I think that "fiddly" gets used as a catch-all term for bad complexity. Obviously complexity is what makes games interesting, but it can also make games unfun if misused.
Types of complexity:
Complexity of the board state. This is measured by the amount of information you have to hold in your head at one time in order to play. Every part of the game in play contributes to board state complexity, but some things contribute a great deal more than other things. Using SotM as an example, abilities like damage boosts, damage reductions, and redirections contribute a great deal to board state complexity. On the other hand, start of turn abilities contribute much less, because you can give the card a threat level and then you don't have to worry about the specifics of the ability until it comes time for the ability to trigger. Also, measuring the complexity by how much information you have to hold in your head means that how card attributes are shown also affects complexity. The ability icons in the video game are reducing the complexity. Also, a character that currently has 29 HP is slightly more complex than a character with 30 HP, because 29 requires more HP tokens to show. This is why Eurogames love using symbols instead of words. Because they are much more visible than words, they reduce the complexity of the board state. (And if you're playing in a foreign language, they reduce the complexity significantly.) Board state complexity is undesirable because it increases player fatigue, but it is necessary to some degree to allow for complex decisions. Which brings us to the next type of complexity
Complexity of decisions. Measured by the number of different options you have at any one time. This is the meat and potatoes of what players who enjoy complex games are looking for. It becomes a problem when the complexity of the action starts to outway the benefit of the optimization. Who here has rewound Argent Adept's power phase when they realized they forgot to order his insturments optimally, and thus won't be gaining 2 HP from Inspiring Supertonic? Or very carefully worked out how to kill a Mountainous Carapace before killing any of Akash's other limbs so as not to miss out on the 1 or 2 points of damage it would absorb? Sometimes you do a lot of work for very little benefit.
Complexity of events. When you deal damage to an Explosives Wagon, which then damages Cryobot, which in turn kills a Sonic Mine, which destroys a Bee Bot, which kills the other Explosives Wagon... More than that, though, is when a simple action causes a lot of alterations to the board state. Tempest does 1 damage to 11 non-hero targets, but only 2 of them have 1 HP tokens on them, so instead of just removing 11 tokens you remove 2 tokens, and then switch out 9 fives for 36 ones. The amount of time spent altering the play area rather than making decisions. If someone is using "fiddly" to mean one specific thing, it is probably this, I think.
There are probably other types of complexity/nuances I've missed, of course.
OH GOD WHERE AM I
To me, “fiddly” refers to the accounting aspects of a game rather than the game’s complexity. I think the best way to characterize “accounting aspects” is, “stuff the player has to mentally keep track of because the game itself doesn’t.” To give an example, Go is extremely complex because there are a vast number of possible decisions and an even more vast number of possible consequences of those decisions - but the game is in no way fiddly, because the board state is 100% captured just by looking at it.
In the example of Sentinels of the Multiverse, I think there are fiddly decks. The Agent Adept is fiddly (did I use the extra power from Inspiring Supertonic yet, or was that from Alacritous Subdominant?). Absolute Zero is often fiddly (Now I’m going to do Thermal Shockwave’s four fire damage to myself - oh, crap, I forgot about this Cold Snap!). Madam Mittermeir’s is fiddly. Haka, Ra, Tachyon, Insula Primalis, and Warlord Voss are not fiddly. They play, (power, draw,) and then just go down the line with end of turn effects.
Ah, I like that expanding of the term, Trajector.
In those terms, I would say that Spirit Island has a little bit of fiddliness, but not nearly as much as SotM, as explained in more detail above.
I just wanted to put this aspect out there. Not to take away from the discussion, but to see how folks interpret/integrate it. 
(I think the definition supports Trajector's thoughts.)
Dictionaries are not necessarily helpful; they can and do miss things. But let's see:
Definition of fiddlychiefly British: requiring close attention to detail : fussy; especially : requiring an annoying amount of close attention