Though many modern resources point to Galentine being named thus due to it's connection with jelly, many early recipes for the sauce do support a connection with Galingale and later ones with at least Ginger which is somewhat close in nature to the root.
Recently I re- stumbled upon these three recipes:
(from Forme of Cury)
"LAUMPROUNS IN GALYNTYNE
Take Lamprouns and scalde hem. seeþ hem, meng powdour galyngale and some of the broth togyder & boile it & do þerto powdour of gyngur & salt. take the Laumprouns & boile hem & lay hem in dysshes. & lay the sewe above. & serue fort."
"SOWPER OF GALYNTYNE
Take powdour of galyngale with sugur and salt and boile it yfere. take brede ytosted. and lay the sewe onoward. and serue it forth."
Take crustes of Brede and grynde hem smale, do þerto powdour of galyngale, of canel, of gyngyner and salt it, tempre it with vynegur and drawe it up þurgh a straynour & messe it forth."
One thing that was not hard to notice is high presence of Galingale in all of these recipes and it would not be such a stretch to see how "Galyntyne" could have connections with "Galyngale" no matter how much we would like it to be connected with "gelatine"...
Another interesting tidbit, from the same collection of recipes, is this sauce quite seemingly named for it's major ingredient/flavouring:
Take payndemayn and pare it clene and funde it in Vinegur, grynde it and temper it wiþ Vynegur, and with powdour gyngur and salt, drawe it thurgh a straynour. and serue forth."
...but this is something we simply have to draw our own conclusions about.
(though one thing to note: not all recipes for this sauce will make any more of a jelly than other sauces, as seen here, though those with protein based broths should at least do that much)