Military Culture, Training & Determination, Dragons & Magic
You have asked "Is there a description of the tactics the Gith employ against Mind Flayers that would allow them to all but exterminate them?"
The 2e game accessory The Illithiad is a book entirely on illithids, their origins, anatomy, culture, future and past history. Regarding the gith uprising, it does not explain specifically how it became successful, though the sheer number of "thralls" who rose up against the illithids was probably an important effect. (The book has an example illithid settlement of 200 illithids and 500 thralls.) It is worth noting that "we" still (5e) do not know how exactly the uprising really turned out to be successful, according to Mike Mearls, who discussed the gith in a D&D Beyond video on December 14, 2017.
According to the book, both the githyanki and githzerai form hunting parties to seek and kill illithids.
- Being part of such a hunting party is part of githyanki culture, every githyanki warrior is expected to serve time. Typical party consists of a githyanki knight armed with a silver sword, a githyanki spellcaster, and nine warriors. About one of three such parties is accompanied by one or two red dragons.
- The githzerai hunting parties (called rrakkma) are not described in detail in Illithiad, but we read about a special psionic power called "mind rip". This power allows githzerai to permanently rip one psionic ability from an opponent. [The 5e Monster Manual (MM) does not mention such a power (as psionics are no longer a separate thing in 5e), but the hunting parties are discussed on page 159.]
So it looks like determination (according to Mearls, the entire githyanki culture is built on fighting the illithids), higher numbers, training and good organization are the keys to the success of the gith, aided by magic (specialized weapons and powers) and red dragons.
The Deal with Tiamat - Comparison of the 5e vs. the older lore
It is quite clear that the aid from red dragons is an important and ancient one. So for the current state of affairs, it would be fair to say that the githyanki are effective against the illithids partially thanks to that alliance.
Regarding the exact starting point of this alliance, there is some conflict in the lore. Towards the end (around 8:07) of the D&D Beyond video, Mike Mearls states that the gith led by Gith did not have help from Tiamat and the red dragons as they initially rose up against the illithids. Tiamat was imprisoned in hell at the time and the deal with her was made after the gith defeated their masters.
What Mearls states in the video is in line with the pre-5e lore. There is a 3e adventure by Chris Perkins, named The Lich-Queen's Beloved, in issue #100 of the Dungeon magazine (July 2003). On page 97, we read:
Years after the githyanki claimed the Astral Plane as their home, Vlaakith I began advising Gith, the githyanki liberator and ruler, in matters of state. In the pursuit of powerful allies, Vlaakith advised Gith to descend into the Nine Hells and meet with the legendary Tiamat, queen of evil dragons.
By the way, this adventure also describes duthka'giths, who are brutal githyanki that carry red dragon blood; they could also be considered as effective opponents against the mind flayers.
Yet, this past lore seems to have been retconned in 5e. First, we have the following sentence from 5e MM (page 159):
In the uprising against the illithids, Gith sought allies. Her adviser Vlaakith appealed to Tiamat, the goddess of evil dragonkind, and Gith ventured into the Nine Hells to meet with her.
And more recently, we read in Mordenkainen's Tome of Foes (page 86):
During the war with the illithids, Vlaakith urged Gith to seek out allies from among the planes and in particular advised her to seek counsel with Tiamat. Gith agreed to venture into the Nine Hells to forge an alliance with the Queen of Dragons. She didn't return.
After the gith overthrew the mind flayers and Zerthimon's followers began to emerge as a threat to Gith's preeminence, Vlaakith played a critical role in ensuring that the githyanki under her rule were protected from an immediate, direct assault by their kin. Using her mastery of arcane magic, she helped the githyanki establish a permanent stronghold on the Astral Plane.