What is the origin of reasonableness?
In other words, what makes a statement sound reasonable? — Isaac Shmukler

Presumably, that the statement has been arrived at by valid logical mechanisms and derives from true premises. In other words, is the product of a sound argument.

This means that a statement is not in itself reasonable, but rather reasonableness is conferred upon it by the means by which it is derived.

What is the origin of reasonableness? — Isaac Shmukler

A reasonable bedtime.

Obviously logic and mathematics have statements that are more than reasonable. Furthermore, their negation is even inconceivable: x+y=y+x, if the object is round, it is not square and so on.
My interest was in the other types of statement: Rich people are more stressed in their lives. Reading more than 6 hours straight is damaging one's vision and so on (these statements do not belong to the "more than reasonable" area and their negation is not inconceivable).

My interest was in the other types of statement: Rich people are more stressed in their lives. Reading more than 6 hours straight is damaging one's vision and so on (these statements do not belong to the "more than reasonable" area and their negation is not inconceivable). — Isaac Shmukler

So we have deductive, inductive and abductive reasoning. The fact, that not everything is as clear cut as your a priori examples does not mean that reason cannot be applied to other areas. Information is reasonable to the extent that reason has been applied in its derivation, but this doesn't mean that it's true.

In other words, what makes a statement sound reasonable? — Isaac Shmukler

Maybe it is something which is delivered in a calm, therefore reasoned, manner.

It implies to me that a state of reasonableness, even though it may state a specific position, is delivered in such a way that it is inviting of other or opposing viewpoints - and that the person who’s position is being discussed, therefore, appears open to having their mind changed (even if this is not what eventuates, ad it depends on what is presented).

re-edit:

.....even if this is not what MAY eventuate, as it depends on what is presented.