This question has been discussed at length here,with a few (in- and out-universe) explanations, but the simplest answer is: he didn't want to free her. This answer is given as one explanation in the above link, but I'd like to expound on it here. Follwing is a simple break down of two intertwined (and somewhat confusing) wagers that, together, ultimately bring about the downfall of the Old Republic. An extensive amount of knowledge of the Star Wars universe is pre-supposed for anyone reading this; I'm not going to explain every detail or situation. Besides, no one without said knowlege would give a rat's ass about this topic anyway.
Almost certainly the main reason Qui-Gon Jinn didn't free Shmi Skywalker was because he wanted to train her son Anakin as a Jedi, and the Jedi were forbidden to have any attachments to family members as they needed to focus solely on their training. When this rule was violated, the results tended to be catastrophic. (See Count Dooku aka Darth Tyranus, in addition to Anakin Skywalker. There are almost certainly other Canon and Legends examples that I'm missing.)
We know Qui-Gon had the means to free her, or at least make the effort, even at the time they were originally on Tatooine. The terms of the initial wager he made with Watto were as follows: Watto supplied the entry fee and the pilot (Anakin,) and Qui-Gon supplied the pod (a great bit of subterfuge as the pod was already Watto's to begin with; presumably as the slaveowner he also owned any property of the slave, in this case Anakin's 'fastest pod ever built.')
If Anakin lost, Watto was to get the Nubian starship (which really wasn't Qui-Gon's to bet in the first place, as he didn't own it). If Anakin won, then Qui-gon got the hyperdrive needed to fix the ship. (Watto was to keep the rest of the winnings he would've made by betting on Anakin, hence Qui-Gon's 'either way you win,' but since Watto bet on Sebulba he lost out big time.)
When Qui-Gon was attempting to free both slaves in his second wager with Watto, the terms of this wager were: Anakin wins, he's free. Anakin loses, he's not free, and Watto also keeps the pod. Again, slick trick, since the pod was already Watto's to begin with.
At first Qui-Gon makes a cursory attempt to free both Shmi and Anakin, but Watto's 'no pod is worth two slaves' put an end to that. Once again Watto's greed gets the better of him; ostensibly the question as to which slave will be freed should Anakin win is left up to chance, but of course Qui-Gon uses the force to settle that question by literally influencing the roll of the die, and it is Anakin who is to be freed if Qui-Gon wins this second wager. Both wagers hinge on whether Anakin wins or loses the Boonta Eve race.
Maybe no pod is worth two slaves, but clearly it is worth one slave to Watto, or he never would have accepted Qui-Gon's second wager at all, and couldn't have been influenced to do so since Toydarians were immune to the old Jed mind trick. The key point left unspoken (or at least unacknowledged in-universe), and the key to the hypothesis that Qui-Gon never wanted Shmi to be freed, was that after all was said and done, in addition to freeing Anakin and getting the hyperdrive components needed to repair his ship, Qui-Gon still owned the 'fastest pod ever built,' at least as far as Watto knew.
Why didn't Qui-Gon attempt to trade the pod to Watto for Shmi? In neither the film nor the novelization of The Phantom Menace is it ever stated or implied that he did so. From the book we know that he sold the pod (most likely to Sebulba) and gave the money to Anakin, who gave it to his mother. The book states that the money from selling the pod was not nearly enough to by Shmi's freedom, but the pod itself, and its potential to earn future money for Watto, might have been a different inducement entirely. Watto just might have been willing to part with his one remaining slave in order to take possession of this incredible pod, which had just defeated the galaxy's premier pod racer (and his presumably top-notch pod) in Sebulba.
This brings up an ethical conundrum, because clearly Qui-Gon Jinn had a vested interest in seeing that Anakin Skywalker's mother remained in bondage (see paragraph 2). He may have been fooling himself by his half-hearted second wager with Watto, but the fact remains that had Shmi accompanied them off Tattooine, Anakin would almost certainly never have been trained as a Jedi. This isn't the only ethical problem that Qui-Gon's behavior presented; for one thing he was gambling with items that weren't his (the Nubian starship belonged to the government of Naboo, the pod to Anakin and hence Watto.) All very interesting behavior for a member of an order that was supposed to represent peace and justice in the galaxy. Here, he flirted with both thievery and slavery.
Supposedly the great Jedi Order knows best in separating children from their families at such a young age they will remember little to nothing about their families: those ties are a distraction, and clearly the worry is that such strong attachments form too many avenues, too many inducements to start down the dark path. Presumably Qui-Gon, in his nascent belief that Anakin might be the chosen one, was attempting to prevent such entanglements for the soon-to-be Padawan. He believes that Anakin will be the one to bring balance to the Force. Ultimately this feels like a very Sith-like proposition, which can be boiled down, simply, to the ends justifying the means.
Fat lot of good that did him; in the end it was Shmi's death that finally pushed Anakin over the edge and firmly, irrevocably down the dark path when he went on his genocidal revenge rampage against the Tuskens who tortured his mother to death. (Let's leave alone for a minute the fact that Padme still loved Anakin even after he admitted to being a mass child-murderer. WTF?)
So ultimately Qui-Gon might have been better off freeing Shmi in the first place: he didn't because he knew that would've rendered impossible the already daunting task of getting the Jedi Council to accept Anakin as a Padawan.
So did Qui-Gon Jinn's two wagers ultimately set off the dominoes that led to the downfall of the Republic and the birth of the Empire? The argument can be made. Hopefully I just made it.
