This is Rabelais, so yes, you can expect a lot of meaning that is not apparent to the casual reader, especially the casual modern reader. Some names are classic references (you've probably spotted Goliath, Polyphemus, Sisyphus…), others are allusions to more recent authors, others are puns, others are purely made up to sound in a certain way.
My answer is based on the Pléiade edition with commentary by Mireille Huchon. Almost every name in this list has a footnote. I'll first translate the footnote regarding the list in general.
This mention summarizes the genealogy and leads to a long development. Biblical and mythological names names alternate with characters from medieval novels and crafted names with obscene allusions. See Officina by Ravisius Textor, 1532, f° 142 r°, which enumerates a list of giants and tall men: (…). From Atlas to Gemmagog, all the names cited here are present in Ravisius Textor's writings, except for Eryx and Cace (two giants killed by Hercules), Enay, Sisyphus and the Titans.
As indicated in A. Lefranc, “La Généalogie de Pantagruel”, Revue des études rabelaisiennes, V, 1907, p. 194, the Titans are mentioned in the commentary on the list, Cacus in the list of “fortissimi homines”, Eryx in the list of Athletae, Sisyphe in the list of dwarfs. The mention of Giants is based on Genesis, VI, 4: according to the Vulgate, Giants are the issue of sons of God and daughters of men, the former being considered in the Middle-Ages to be descendants of Abel, the latter of Cain. In Ovid's writings as well (Metamorphoses, I, 135 ff.) as well, the earth, impregnated by the blood of giants, produces a new race of men (see E. M. Duval, The Design of Rabelais's Pantagruel, p. 22).
Even without any reference material, there's a visible progression: first Hebrew-sounding names, then Greek names, then Latin, then mixed origins for a while, then old French, then modern French. So the order follows the expected cultural references for the main periods of history in chronological order.
Here's some anecdotal information about some of the names, based on Mireille Huchon's footnotes. I picked footnotes that I considered to be interesting, this is not a statistically significant sample.
- Nembroth (known today as Nimrod) is cited in Boccacio's Genealogy of the Gods as one of the two Biblical giants who were instrumental in building the Tower of Babel. The other -broth names are made up.
- Hurtaly is based on Og according to [M. A. Screech]'s edition of Rabelais: this is a parody of a tale by Rabbi Eliezar (I don't know which of the several famous rabbis whose name this could be a transliteration of). Og sat on the ladder outside Noah's arch and Noah passed him food through a hole. According to Lazăr Șăineanu, La Langue de Rabelais, 1922, II, p. 35, the name is a deformation of the Hebrew Ha-palit, meaning “he who escaped”, based on a Biblical commentary that Rabelais is known to have read.
- Eryx is the son of Venus and either Butes (mentioned Virgil's Aeneid) or Neptune. He fought with enormous gloves made of ox skin with iron barbs which he threw at his opponents.
- Etion is cited by Ravisius Textor. His body was supposedly unearthed in Crete after an earthquake. Giovanni Bertachini is an Italian legal scholar whom Rabelais didn't like. I have no information about his drinking habits or illnesses.
- Encelade, Cée, Typhoe, Aloe (Alous), Athlas, Briaré are sons of Titan. Briareus has a hundred hands in Greek mythology.
- Gabbara is mentioned in Pliny and Ravisius Textor. Offot is mentioned in Ravisius Textor. I have no information on their drinking habits.
- Happe mousche is still recognizable in modern French as meaning “fly-swallower”.
- Bolivorax is a Greek coinage meaning “earth-eater”.
- Gayoffe is the name a character in Teofilo Folengo's Baldo, which is known to have inspired Rabelais. I don't know if the characteristics attributed to him by Rabelais are from Folengo.
- Galehault (at least the name) is from Arthurian tales.
- Galaffre, Sortibrant and Mabrun and a few others are the names of Saracen kings, taken from the Song of Roland or other Charlemagne legends.
- Foutasnlon (Foutasnon in my edition) sounds obscene. I can't fully explain the name, especially given the two variant spellings, but the beginning fout- means “fuck”. According to Roland Antonioli, Rabelais et la médecine, p. 40, this may be an allusion to a medical doctor from Montpellier called Denis Fontanon.
- Vitdegrain means “grain-dick”. (I don't know what “grain” connotes here, if anything.)