This is from The Autobiography and Correspondence of Mary Granville, Mrs. Delany edited by Lady Llanofer and published in 1861-62. Mary Granville Delaney (1700-1788) became famous in her 70s for her decoupage creations some of which are apparently preserved in the British Museum. Her autobiography is available at Google Books (the above paragraph is from p. 477).
Could there be something in boiled snail extract, perhaps something coming from snail slime, that could suppress coughs? Presumably, Mary Delaney was referring to some common and large British land snails. Cepaea or Helix, maybe?