1. I note that you say that when N = 3, there's no new information given. But when N = 2, all islanders are still aware that there is at least one person with blue eyes.
I'm guessing that despite that fact, you see how in the N = 2 case, the two people with blue eyes can infer they have blue eyes. So the leap of intuition you should've had there is that somehow, new information is being conveyed, even though every islander already knows that there is at least one person with blue eyes.
2. But let's just tackle this with straightforward inductive logic. For now, we aren't going to worry about what new information is given, we're just going to do induction.
If I see that N people have blue eyes, then I know that either there are N people with blue eyes, or there are N+1 people with blue eyes, and I have blue eyes. If after N days, the people I see with blue eyes haven't killed themselves, then I know that there must be N+1 people with blue eyes, and I have blue eyes, ergo I must kill myself.
This holds with N=0 and N=1 and N=2, so it on some level doesn't matter what the logic is, induction works.
3. But okay, let's figure out what information is actually given. Let's say that on a 100 person island, I have blue eyes and Alice and Bob also have blue eyes. So from my perspective, I know that there are two possible scenarios:
a. Alice, Bob, and I have blue eyes. b. Only Alice and Bob have blue eyes.
Let's consider situation b. In situation b, what does Alice think? Since we're stipulating that I have brown eyes, Alice sees that Bob has blue eyes and that's it. Therefor, Alice thinks that there are two subscenarios of situation b:
b1: Alice and Bob have blue eyes. b2: Only Bob has blue eyes.
Hypothetical Alice in hypothetical situation b considers situation b2. She thinks about what Bob would know in situation b2. He would see that nobody besides potentially him has blue eyes. So he would kill himself when the visitor explained that at least one person had blue eyes.
If Bob does not kill himself, then hypothetical Alice in situation b knows that she is not in situation b2. There is only one other subscenario of situation b, and that is b1, in which both Alice and Bob have blue eyes. So Alice would kill herself on day 2 (as would Bob).
If Alice does not do that, then Alice clearly did not consider herself to be in situation b. The only alternative to situation b is situation a, in which Alice, Bob, and I have blue eyes. So we all kill ourselves on day 3.
And obviously since all of this is symmetrical -- neither Alice, Bob, or I have any privileged information, that means that if I can see three people with blue eyes, and they don't kill themselves on day 3, then the only other valid scenario is that I also have blue eyes, so on day 4, I kill myself.
The information added is information about what we know that other islanders know about people's eyes, not what we know about people's eyes.