Possessives – Can One Say ‘House’s Roof’ or Only ‘Roof of the House’?

I think you could say it either way, but you'd want to switch articles:

A human's brain works better in the morning.
The human brain works better in the morning.

The latter uses "the human brain" in the general sense, as @MaulikV explains in his answer. The former talks about a person's brain in the indefinite sense.

I think the latter would be more appropriate for scientific contexts, yet the former is acceptable in casual conversation.

The format with the indefinite article also requires a possessive; consider:

The strongest muscle in the body is a leg.
The strongest muscle in the body is the leg.

Here, the indefinite article doesn't sound right. However, we can fix that with a possessive modifier:

The strongest muscle in the body is a person's leg.

Old English (for the sake of simplicity, English before the Norman Conquest in the year 1066) had declensions, like German. The feminine third person singular genitive and dative forms of the pronoun were the same (hire) whereas the masculine forms were his and him.