That tight hair of hern?

Yes. If she'd just let that tight hair of hern all out loose and careless-like, as it used ter be, and wear the sort of bunnits with posies in 'em, and the kind o' dresses all lace and white things--you'd see she'd be handsome! Miss Polly ain't old, Nancy.

(Eleanor H. Porter, Polyanna)

In the above paragraph, what does the “tight hair of hern” mean? A hern is a bird, I think. Or does it have another meaning?

You are right in saying that hern can refer to a bird—a heron in modern spelling, but spelled without the o archaically and (according to Merriam-Webster's Eleventh Collegiate Dictionary) in at least one dialect of English today. One famous example of this usage appears in Alfred Tennyson's poem "The Brook" (1864):

I come from the haunts of coot and hern,

I make a sudden sally,

And sparkle out among the fern,

To bicker down a valley.

Not only did the words of this poem pass before many a schoolchild's glazèd eye in many a hush'd classroom, but the opening line served as the caption of a memorable cartoon by James Thurber.

However, as Edwin Ashworth points out in his comment above, the hern in your quotation isn't a bird—or a noun—at all. It's a pronoun, a variant of the standard pronoun hers, and it makes a pair with the masculine form hisn, and a whole family with mine, thine, ourn, yourn, and theirn added. Unfortunately, the Eleventh Collegiate doesn't acknowledge most of these words' existence, but according to a writer in Language & Communication (1980), they derive from the model of mine:

The analogizing of the /-n/ in the pronoun mine to other pronoun forms yields ourn, yourn, hern, hisn, and theirn.

Samuel Pegge, Anecdotes of the English Language; Chiefly Regarding the Local Dialect of London and Its Environs (first published in 1803 after the author's death) notes the use of hern and related pronoun forms, and devotes an erudite chapter to the subject. Pegge rejects the then-current folk etymology claim that all of these forms represented contractions (of "my own," "thy own," "our own," "your own," etc.), observing that mine, thine, ourn, and yourn are all "actual Saxon pronouns possessive." A footnote to that chapter finds an example of hisn (albeit spelled hizzen) from 1575:

In the year 1575, Master R. Laneham, who seems to have been a Keeper of the Council Chamber, and a travelled man (though perhaps by birth and breeding a Cockney), writes to his friend Master Humphrey Marti, a mercer, an account of Queen Elizabeth's reception and enteryainment at Kenilworth Castle, wherein he describes some person who, after praying for her Mjecty's perpetual felicity, finishes with the humblest subjection both of "him and hizzen."

Because using the hisn/hern family of pronouns was considered low and vulgar, few educated writers resorted to these forms. But in the 1800s in the United States, as dialect speech became popular in fiction, the forms began to show up. For example, Harriet Beecher Stowe, Uncle Tom's Cabin (1852) uses hern twice within the space of two pages:

Thought so!—didn't she Thar she was, showing 'em, as innocent—ye see, it's jest here, Jinny don't know. Lor, the family an't nothing! She can't be spected to know. Ta'nt no fault o' hern. Ah, Mas'r George, you doesn't know half your privileges in yer family and bringin' up!"

...

Say?—why, she kinder larfed in her eyes—dem great handsome eyes o' hern; and, says she, 'Well, Aunt Chloe, I think you are about in the right on 't.' says she; and she went off in de parlor. She oughter cracked me over de head for bein' so sarcy; but dar's whar 't is—I can't do nothin' with ladies in de kitchen!"

And of course hern was still in dialect use in 1913 when Eleanor Hodgman Porter's Pollyanna (the source of your quotation) appeared.