Meaning of "They say! What say they? Let them say."

Solution 1:

This would be in response to a reproach grounded in folk wisdom or popular opinion—perhaps interrupting the reproach after just two words, the two spat back at the beginning of this “motto.” To paraphrase:

You throw popular opinion in my face? What is that popular opinion? No, never mind—let the great mass of fools say whatever they will.

Solution 2:

SUPPLEMENTAL to Brian Donovan's ANSWER:

The slogan was well known in the later 19th and early 20th century: Lily Langtry, for instance, is said to have had it either carved over her mantle or displayed in stained glass in the house the Prince of Wales built for her, and Bernard Shaw found it carved over the mantle of the house he bought in Adelphi Terrace.

The unusual syntax may be attributed to an origin in Renaissance Scots; it appears to have been originally the motto of Marischal College in Aberdeen, founded by the Earl Marischal of Scotland George Keith in 1593. A number of internet sources cite the oldest version as THAI HAIF SAID : QUHAT SAY THAY : LAT THAME SAY. But the earliest actual citation I have found is from 1824, in one of John MacCulloch's 'letters' to Walter Scott in The Highlands and Western Isles....

There are several posts in Notes & Queries claiming classical precedents, Latin (Aiunt. Quid aiunt? Aiant.) and Greek (λέγουσιν ἄ θέγουσιν λεγέτωσαν), and Keith was said to be a fine scholar in these languages.