Learn English – Where did the practice of using apostrophes for possessive nouns but not pronouns originate

Professor David Crystal explains it in his book The Fight for English: How language pundits ate, shot, and left (Crystal 2006), pp. 134-135:

Its is just as possessive as cat's, but it doesn't have an apostrophe. Why not? Because the printers and grammarians [of the nineteenth century - Alex B.] never thought the matter through [emphasis mine - Alex B.]. They applied their rule to nouns and forgot about pronouns, thus creating an exception (along with the food is hers, ours, yours, theirs) without realizing it. And even if they had noticed, they wouldn't have done anything about it, for it's was already taken, as it were, as the abbreviation of it is.

I highly recommend this book to anyone interested in languages. 

Charles Fries (Fries 1927) speculates that it could also be so because "their plural forms do not end in s" (cf. one - one's - ones or other - other's - others, ft. 7)

For an excellent summary of how the rules concerning apostrophe use developed, see

Sklar, E. (1976). The Possessive Apostrophe: The Development and Decline of a Crooked Mark. College English, 38(2), 175-183. doi:10.2307/376342

If you want to learn more about how the grammarians of the past arbitrarily imposed their confusing rules - and didn't stick to them - see pages 197-198 in Doctrine of correctness in English usage, 1700-1800 by S.A. Leonard (Leonard 1962); for instance, Joseph Priestly argued in The rudiments of English grammar, which was published in 1772, the following (pp. 86-87):

On the other hand, in the same book, on page 11, he lists all the possessive pronouns without an apostrophe and he treats its separately, as the genitive form:

Thirteen years later, J. Mennye in An English grammar ; being a compilation from the works of such grammarians as have acquired the approbation of the public [...] argued diametrically the opposite of the convention earlier proposed by Joseph Priestly.

But in 1823, T.O. Churchill says the following in A New Grammar of the English Language:

This is perhaps best explained by providing the relevant extract from the ‘Longman Student Grammar of Spoken and Written English’:

The double genitive is a special construction in which either the independent genitive or a possessive pronoun occurs in an of-phrase:

This was a good idea of Johnny’s.

There’s a talk by this lady from Boulder who’s a student of Sandy’s.

The woman who owns Harte’s is a friend of ours.

As these examples show, the main noun phrase typically begins with the indefinite article. In fact, the definite article does not normally combine with the double genitive: *the good idea of Johnny’s is unlikely to occur.

The meaning of the double genitive can sometimes be alternatively expressed by other constructions. Thus, a friend of ours could alternatively be expressed as one of our friends.

Here is a further explanation from ‘The Cambridge Guide to English Usage’:

The double genitive seems to serve two purposes:

emphasis. This is the effect of paraphrasing “not Jo’s fault" as no fault of Jo’s, or turning “our friend” into a friend of ours. The double genitive unpacks the phrase and foregrounds the noun rather than the person. In conversational examples such as That book of Bill Bryson’s is his best yet, the construction helps to adjust the topical focus.

clarification. Clearly a painting of Lady Rich’s and a painting of Lady Rich mean different things. The first (a possessive) makes the painting part of Lady Rich’s collection, while the second (technically an objective genitive) says that it is a portrait of the Lady herself. The duplication of the genitive is thus not redundant but clarifies the fact that the first construction is a possessive genitive.