Is there a concept of information that distinguishes between "computed" and "non-computed" information? I always tend to think of information as follows: If we have an information set $I$, and it is possible to derive from $I$ another information set $X$, then $I$ really contained $X$ all along, and we haven't actually discovered "new information". We've merely found out something new about the information we already had.

But sometimes, deriving $X$ from $I$ is computationally very complex, so that it may take 1000 years to do it. In this case, any reasonable person in every day life, would say that we don't have the information $X$, even though we have $I$ and $X$ can theoretically be computed from $I$. 

My question is, is there a concept that captures this notion of not having a piece of information merely because it is hard to compute? i.e. it could be something like: $I_{405}^*$ is the information set containing all information that can be generated by at most 405 computational steps from $I$. This is a somewhat vague definition, but I'm wondering if there is something more formal
The distinction between information theoretic and computational knowledge is an issue which is fairly common in cryptography.

Consider for example the one-time-pad encryption, that given a message $m\in\{0,1\}^l$, encrypts it using a secret key $k$ (chosen uniformly at random) by $m\oplus k$. Clearly, one learns no information from seeing the encryption of one message (it is a uniformly distributed string), so you could say that you have information theoretic security, which should make you very happy. However, if you insist on such a strict notion of security, it turns out you cant encrypt messages who are longer than the key, and if you want to do that, you should settle for a weaker notion, which is security against bounded resources adversaries (see semantic security).

One example which seems to fit your question well is one way permutations. A function $f:\{0,1\}^*\rightarrow \{0,1\}^*$ is called a one-way permutation if for every $n\in\mathbb{N}$, $f|_{\{0,1\}^n}$ is a permutation of $\{0,1\}^n$, and for some bounded resources adversaries, almost no knowledge of $x$ is exposed from $f(x)$. Since $f$ is a permutation, $f(x)$ completely determines $x$, but it is extremely hard to recover.

That said, I will strengthen Raphel's comment and say that there isn't a general definition for information (or what you call computational information), this is an issue which strongly depends on the context.