Are there any old third-person singular pronouns the way “thou” is an old second-person pronoun?

Thou is an old second-person singular pronoun in English.

Are there any old third-person singular pronouns?

Personal Pronouns: Rustic hisn and hern, hit and sho

Because Old English and Middle English each had their own sets of pronouns that differ from ours, let’s restrict this to Modern English only, not anything earlier like The Canterbury Tales or Sir Gawain and the Green Knight, respectively from two different branches of Middle English.

Otherwise we’d have to discuss things like the pronoun hit that we now spell without the leading h- these days outside of various regional dialects. English dialects do preserve a tremendous wealth of such ancient forms that are still used in dialect speech. For example:

• 1993 New Shetlander Sept. 29
  Hit’s a notion shu’s taen lately.

There hit means it and shu (or sho) means she. There are many, many more of these that you come across from time to time.

Even so, we can still find other archaic personal pronouns that persist today only in dialect, not in standard English. The most notable of these is probably the -n genitives formed by analogy with mine and thine, including hisn, hern, theirn, ourn, yourn. The one that goes with his has many spellings, being non-standard. These include hisn, his’n, hisen, hissen, hizzen.

The paywalled OED gives as the etymology of hisn:

Alteration of his adj. after mine pron., thine pron. Compare hern pron.1, hern pron.2, theirn pron., ourn pron., yourn pron., and compare also hisis pron.

Eng. Dial. Dict. records the word from Yorkshire and Lancashire and widely from the midlands and south of England; Surv. Eng. Dial. records the word from the majority of midland and southern counties, with the exception of Cheshire to the north, Rutland, Norfolk, Kent, and Sussex to the east, and Dorset, Devon, and Cornwall to the west. Dict. Amer. Regional Eng. records the word chiefly from New England and southern and south Midland states.

Compare also English and U.S. regional hisn’s, pronoun and adjective (19th cent.).

These started in Late Middle English, and made the transition into Early Modern English. However, they were never really considered part of standard English, and modern writers have used them only to represent the speech characteristic of various regional dialects, where they still exist to this day. The OED says that hisn is:

Now English regional (midlands and southern) and U.S. regional (chiefly south Midland, southern, and New England).

They give genitive examples corresponding to mine and thine, including these later ones:

• 1867 ‘M. Twain’ Celebrated Jumping Frog 15
  It always makes me feel sorry when I think of that last fight of his’n.
• 1930 W. Faulkner As I lay Dying 115
  He don’t say nothing; just looks at me with them queer eyes of hisn that makes folks talk.
• 1938 W. E. Railey Hist. Woodford County ii. 55/1
  I trust the old mill will be to ‘him and hisn’ a sacred trust.
• 1946 L. Lenski Blue Ridge Billy viii. 108
  I’ve got more baskets than I can sell in a month of Sundays. Uncle Pozy come by here t’other day..and wished all his’n on me.
• 1983 E. McClanahan Nat. Man (1984) iii. 22
  Old Nobs thinks this new boy of his’n is gonna be shit on a stick, don’t he?
• 2011 S. Rodriguez Weight of Light iii. 36
  My tongue feel thick and swole like hisn, and I can’t speak.

They also give possessive examples corresponding to my and thy of which these are just the more recent:

• 1910 C. E. Mulford Hopalong Cassidy xx. 130
  Taking the button and looking it over. ‘Yep, it’s hissn, all right.’
• 1923 ‘R. Crompton’ William Again x. 179
  Well it isn’t his’n—it’s stole stuff.
• 1995 M. L. Settle Choices ɪ. vi. 76
  He’s got seven kids, four of them hissen.
• 2004 J. B. Rehder Appalachian Folkways 297
  That plate is hern... But the cup is hisn.

Notice how these examples are almost all meant to represent the rustic speech of regional dialects.

Demonstrative Pronouns: yon, thon, thilk

Another source of archaic pronouns still occasionally heard is the demonstrative pronouns beyond the four standard ones this, that, those, these. These include yon, yond, yonder, thilk, thon.