Why does English employ double possessive pronouns such as theirs and ours?

I don't know why etymonline calls theirs 'a double possessive', but it's not.

The unfortunate terminology of 'the double possessive', aka 'the double genitive', is not due to the pronoun theirs itself but to the common construction like a friend of theirs where traditional grammar treats the preposition of as another possessive on top of the possessive pronoun theirs.

So in a construction that doesn't contain of, theirs itself is no double possessive:

The book is theirs.

This example of yours, for example, doesn't contain of, so there's only one possessive, the possessive pronoun theirs, which means their book. (Note the subject of the first clause does contain of, so you can call it a double possessive.)

Now, some grammarians don't like the term 'the double possessive/genitive' even for constructions like a friend of theirs.

The Cambridge Grammar of the English Language (Page 468), for example, treats She's a friend of Kim's not as a double possessive/genitive but as a oblique genitive:

...we do not regard of as a genitive case marker, and hence there is only one genitive here, not two.

As for the distinction between their and theirs, CGEL classifies the former as a dependent genitive (possessive) and the latter as an independent genitive (possessive), which easily explains why these don't work:

*The book is their.

*That is theirs book.