According to the Wikipedia article on rose Bertin, a milliner and dressmaker for Marie Antoinette, Bertin is the person responsible for the quote. Bertin

is said to have remarked to Marie Antoinette in 1785, when presenting her with a remodelled dress, ” Il n’y a de nouveau que ce qui est oubliu00e9 ” (“There is nothing new except what has been forgotten”).

The Edinburgh Review, Or Critical Journal on the subject of “Education of the Poor in France” cited by Wikipedia! Of which no nation

makes such frequent application as the French, and which, as history relates, was the favourite maxim of the most inventive and academic of dressmakers, Mademoiselle Bertin, is, ‘Il n’y a de nouveau que ce qui est oubliu00e9;’ and we think the history of these didactic inventions affords a striking proof of its justice.

How did a reviewer writing for a Scottish publication in 1820 assert that the saying in question was a French proverb, and one that Marie Antoinette’s dressmaker particularly liked. …? That is probably as close as the proverb’s origin is likely to get.

As Edwin Ashworth notes in an earlier comment, the sense of the saying that what strikes us as new and different is in all likelihood merely old and forgotten—but recently rediscovered as only new. In a world where everything that can be done has been done (such as in late eighteenth-century France, evidently) there is nothing new under the sun, so the best one can do for surprise, excitement, and the appearance of innovation is to resurrect something that no one remembers.