My nephew has picked up the habit of responding, “I know” when given some information. Naturally, at 10, he doesn’t know - most of the time.
How can I best teach him that this is an inappropriate response? I don’t want to belittle him, and giving him an opportunity to strut his stuff when he does know would be ideal.
Seems like he’s just saying that to tell you he’s paying attention when he’s really not interested. Say, “You know, nephew, I noticed you pee your pants at least 12 times a day…” 
Doubt he’ll say “I know,” if he’s listening. If he does, he’s not listening, and he’s just a healthy pre-teen.
Make up Cliff Clavin-style factoids and see if you get the same reaction. “It’s a little known fact that the iPad was actually invented by the ancient Babylonians, only theirs were made out of clay and instead of Angry Birds they played Rabid Camels.”
It’s not an inappropriate response for a 10-year-old. “Fuck you, Uncle Quartz!” is an inappropriate response. Let him be; in a month he’ll be answering with something else. Don’t you remember what it’s like to be ten?
Not important enough to worry about, IMO.
He will soon grow out of it.
Why stop him? Unless you’re as literal as a 40s SF robot, it’s not a problem.
If he’s saying “I know” to statements of fact that he couldn’t possibly know, that seems like an odd thing to do.
But if he’s saying “I know” when you’re giving him pieces of advice, in that case “I know” translates to “Your advice is just common sense and you’re wasting both of our time by trying to lecture me”. At least, in my experience. 
He may actually know or feel he does, perhaps ask him how he knows. Some children are downright intuitive and have a unexplainable gifted grasp of things. IMHO explore deeper why he says it without discouraging him and of wanting him to just stop.
Peace
Instead of telling him stuff ask him what he knows or what he thinks and build on it.
I think it’s probably just a phase, but here’s an idea. The next time he tells you something, say ‘‘I know.’’ He’ll take a hint eventually.
This is often an indication that the information giver is patronizing, or offering unwanted information. So why are you giving your nephew information? How are you doing it?
Or he’s just a kid and it’s an innocuous speech pattern that will be gone a week from now.
I know.
My stepson did this incessantly throughout his teenage years. It was like a catch-all response. Drove me nuts.
“You need to get out of bed and get ready right now!”
“I know!”
“I’m not interested in what you know. Your knowledge, or lack thereof, is not the issue here. What I am concerned about is your action. You KNOW what to do, but you’re not DOING IT! GET OUT OF BED!!”
Indeed. In a couple years, he’ll just mumble something like “Um-hmm”. Enjoy this particular talkative phase he’s going through!
If it’s an issue of manners, in general I think a frontal approach is best. “When you say, ‘I know,’ it comes across as rude and arrogant, and it makes me feel irritated. Even if you truly already know the information someone’s giving you, it’s usually not a very polite response to tell them that. A better response to someone giving you information is ‘Gotcha,’ or ‘Okay, thanks,’ or ‘Cool.’ Would you try using one of those responses instead?”
kanicbird’s point about intuitive kids is spot-on, only not in the way he thinks it is: this kid is not very intuitive about how people respond to him, which is true about a lot of kids. Telling him in a blunt fashion, in a way you wouldn’t do for adults, is part of how you offer him an education.
Why didn’t you start with “Get out of bed!” instead of pointlessly stating some obvious information? 
That’s a great way to elicit a “Fuck you, Uncle Quartz” response.
Seriously, why the need to micromanage your nephews responses like that? If it’s really a problem for you, talk to his parents and let them decide what action to take (or not). Otherwise, make a joke out of it, as suggested by Telperioin. If you let the kid know he’s getting to you, he will have won, and you’ll lose his respect. Just go with it, and don’t try to be such a controlling adult over something that just isn’t that important.
It depends if he’s in something of an authority role for the nephew. If he is, then he’s doing the nephew a favor by making explicit an effect of his behavior: what annoys the uncle almost certainly annoys other folks as well, and kids need to be told both the good and the bad things they’re doing. It’s not micromanaging, it’s educating.
And if the nephew is so unimaginably rude as you suggest, then he needs a lot of help. However, you’re probably just projecting your adult attitude onto a 10-year-old. Kids =/= adults, and the ways you talk to kids =/= the ways you talk to adults, one hopes.
If it annoys you, just start teasing him a little saying “oh do you now” or “I have a genius for a nephew!”.
Eg:
Uncle Q: “Did you know bonobos have a sense of humour?”
Nephew: “I know…”
Uncle Q: “Ooooh do you now? Well aren’t I lucky with a certified genius for a nephew! And what else do you know about bonobos?”