Gamgee Tissue has a thick layer of absorbent cotton wool between two layers of absorbent gauze. It represents the first use of cotton wool in a medical context, and was a major advancement in the prevention of infection of surgical wounds. It is still the basis for many modern surgical dressings. The name is a trademark of Robinson Healthcare Ltd, based in Worksop, Nottinghamshire, UK and has been since 1911.
The connection is not certain: in Appendix F to The Lord of the Rings, Tolkien mentions, but at the same time denies, the reading of Gamgee as a pun relating to the name of Sam's wife, Rosie Cotton. He further elaborates the 'real' Westron names of which 'Gamgee' and 'Cotton' are translations.
However, in the same section Tolkien also addresses hobbit (which was certainly created first and translated afterward, as described in Tolkien's own comment on the initial writing of The Hobbit) and Brandywine (an obvious English pun on the Elvish Baranduin, 'justified' as translation of a similarly alcoholic pun in Westron). In this context we may suspect that the Professor is speaking tongue-in-cheek; the Gamgee entry is phrased such that it can be read as a poker-faced, academic way of pointing out that the joke is there, whether or not he intended it.