[zaptheimpaler]

I think information can not be defined based on the source material alone. Why don't we define information like this?

An observer X has a prior estimate of the distribution d1 of variable x.

The source material yields a distribution d2 of variable x.

Then for that observer, the information gained from the source is something like |d1-d2|.

[Twisol]

Yes, you can think of it like an information game where your function is a black box that you can feed values and get back values. If you take lots of samples from a uniform distribution and feed it to the black box, you can estimate how the function maps one distribution to the other, even though you might not know where all the individual values in the domain go. A good hash function produces a uniform distribution of some variety no matter what distribution you feed it. Bad hash functions deviate from uniform in some way correlating with the input distribution, i.e. they leak information.

The ideal hash function leaks no information. In practice, we can only make one leak very little over a long time.