This topic came about as a result of a panel I sat on for Sasquan (Worldcon 2015) with Denise Connell, Teresa Nielson Hayden, and Robert J. Sawyer. Steven Silver was also going to be on the panel but had to cancel.
I'm not going to recap the panel, because that would involve putting words in the mouths of people that are far more capable of representing themselves with more eloquence than I can muster. I will, however, go over some things that we didn't get to go into in depth due to time constraints.
The moderator suggested that we read this article to prep for the panel. I did and I was glad. It was fascinating. I suggest that you read it as well, but in case you don't have the time (in which case I thank you very much for spending your limited time with me!) the two sentence summary is that authors like Sir Arthur Conan Doyle stack the deck in their character's favor to make them look smarter, and don't insult your audience. The author (Graham Moore, screenwriter for The Imitation Game) also wrote a neat twist at the end of the article.
I write quite a few characters that are smarter than me. They're not all intelligent in the same ways. I'll give you my initial list, and you can build your own from there.
Spacial Intelligence: The ability to visualize in three or more dimensions and to be able to map and rearrange things. These things may be objects that are literally oriented to each other, like shapes in a Tetris game, or they could be scenes in a book, lines or concepts in a poem that's being developed, social maps (how people in a group or how various groups relate to each other) or even more abstract things I can't even imagine. People with spacial intelligence can create metaphors for what they're studying that they can visualize in their minds or on paper and then orient and reorient them to learn new things and to create greater efficiency, artistry or effectiveness in the system they're modeling.
Organizational Intelligence: Less abstract than spacial intelligence, organizational intelligence takes an existing body of data and sorts it or resorts it so that it makes sense or reveals a useful pattern that can lead to problem solving. People who are organized can appear to be extremely intelligent even if they're of average intelligence simply because they can find what they need and can realize when something is missing, duplicated or out-of-place.
Emotional Intelligence: They walk in the room and instantly know who is nervous, who is bursting with news, who is intent on violence and how to manipulate each and every one of these people. They can comfort grieving friends while the rest of us stand around awkwardly not knowing what to say, and figure out how to get the general of the army to stand his men down until news of the treaty arrives. This is a largely overlooked form of intelligence. I use it extensively in The Lord Jester's Legacy and it's also important in The Poisoned Past trilogy.
Fact Collecting: Trivial Pursuit made useful. It's an element of Sherlock Holmes' intelligence, and he's often depicted running various experiments in order to build his mental catalogue of seemingly useless bits of information that later prove critical to his ability to solve crimes. It's a bit of a cliche' that individuals who collect facts tend to rattle them off even if no one cares to hear about them. Although this can be true, it's not a requirement to make your fact collector eager to share his or her body of knowlege at dinner parties or on dates.
Memory: A well-honed and/or a genetically enhanced memory is extremely useful. Being able to recall entire conversations helps catch liars (or sustain lies), and people who have eidetic ('photographic') memories can, for example, picture a page in a book they'd glanced at and reread it to themselves to pick out relevant details long after they no longer have access to it. People with good memories tend to do well academically. They may have an advantage when learning languages as well (though that is a separate skill) and also have an opportunity to do well socially. I had a college professor who memorized the names of every student in his classes every term. He was very popular as a result, except among the people who wanted to skip classes. People who remember the names of your kids and ask whether your aunt's surgery tend to be well-liked.
Perceptiveness: People with perceptive intelligence live more in the present than most people, who tend to live in their heads (or on their iPhones.) They notice things that most people overlook. Some perception is very specialized. My daughter can spot wildlife that's invisible to most people, but she has no idea when someone is flirting with her. Sometimes perception is an ability to get an instant feel for a situation without being able to point specifically at the clues that give that impression. For example, someone might get a bad feeling when they go into a bar without actually noticing that all the people there seem to know each other and that most of them tensed when the stranger came in.
Observing the Negative: A specific, conscious form of perceptiveness. People who observe the negative notice what's missing or what's been deliberately omitted. It might be a ring on a finger that has a pale band on an otherwise tanned hand. Or, when someone remarks that they're paranoid and can't stand to be too close to people, the person who can observe the negative notices that this supposedly paranoid person is voluntarily giving up information without being asked and is standing three feet away. People who are able to detect the negative visually often find the trick behind optical illusions more quickly.
Lateral thinkers: They rarely take things for granted and often look for more options even after a solution or explanation has been agreed upon. Lateral thinkers make a game of figuring out many ways of interpreting data. Rather than going for the first, easiest route, they search for alternatives, either as an exercise or because they realize that choosing the best among many options has great advantages over picking the first workable option, and increases the chances of discovering a compromise or hybrid option to a difficult problem that appears to have only one solution. Lateral thinkers tend to be artistic, creative, and can improvise.
Engineers: Engineers take a physical problem, such as how to cross a river, and come up with a solution with the available materials. They may rely on lateral thinking, but more often they employ a form of spacial intelligence combined with an excellent understanding of practical physics. In my experience, some of the best engineering minds exist among farmers, who have to solve physical problems, such as fixing gates and moving large loads of wood up a steep hill, on a daily basis.
That should get you started.
For additional reading:
A nifty commentary on the Dunning Kruger effect
Psychology of Intelligence Analysis
Lateral Thinking (ebook format)