This is what your player is expecting:
This is not how 5e works (and as a matter of fact, not how most RPG systems I am aware work).
It seems your player is not familiar with the system (it's comprehensible - it's practically their first time playing), and therefore their expectations are completely off from the actual game design.
Talk about these expectations
The system provides a baseline on what the expectations from the players (DM included) should be. That doesn't mean the table can't talk about it and change it if they feel the game will be funnier and more appropriate for them. If you guys can find a middle ground on the expectations for the game, that's the best outcome.
Again: the player should be familiar with the book and the rules. There are a few things that the player is messing up here. First, they are completely stepping on DM's territory trying to dictate how the NPCs should behave. That is DM's job. See this related question about DM and players mixing their roles and turning the gaming experience into a very annoying experience (I can assure you the experience was annoying: the question is mine). So, let us remember the basic foundation of the game, described in Page 6 of the PHB, under How to Play:
- The DM describes the environment.
- The players describe what they want to do.
- The DM narrates the results of the adventurers' actions.
Period. The player influences the world through its actions, and some tables may have the players (the real people) be part of the world construction as a whole (Korvin and Nitsua may help you more with that than me), but, ultimately, in D&D 5e, the DM dictates the results of the players' actions, and other than stating their dissatisfaction, the players should not feel entitled to have anything more than that.
Furthermore, the rules are clear on how ability checks work.
To make an ability check, roll a d20 and add the relevant ability modifier. As with other d20 rolls, apply bonuses and penalties, and compare the total to the DC. If the total equals or exceeds the DC, the ability check is a success — the creature overcomes the challenge at hand. Otherwise, it's a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM.
Now, the problem here is that the DM and the player had a differing view on what a success means. The played succeeding on intimidating the goblins, which means they have succeeded. But for them, this success should imply on yet another consequence: giving away Sildar for free. This is not the action the player described they wanted to do in Step 2 from How to Play - what they described was intimidating the Goblin. If the player wanted to intimidate the goblins in giving away Sildar for free, that is the description they should have given - and then the DM should not even ask for a roll: just tell them that is impossible, the goblins are not willing to do that, no matter how high the roll is.
In particular, as mentioned in other answers, a nat 20 has no special meaning in ability checks, and, in fact, has a very limited special meaning even in attacks - it is a critical hit, but that is all. No limbs being dismembered (at least not by default - you can include that with some DMG optional rules), no instant kill, no extra anything - just extra dice being rolled in the damage. Going back to talk about expectations, you may decide to house-rule otherwise, if the table agrees, and make nat 20s have special meaning. But keep in mind that it works the other way around - NPCs also roll dice and also have nat 20s, and your players may not like when that happens. In my experience, trying to give special meaning to nat 20s and nat 1s in 5e has always resulted in frustration and has shown to be a bad rule, but that is my personal experience and your table may find otherwise.
It could have been gradual
So, after the first intimidation, the goblins decrease the price. The player (and therefore the character) is still not happy. Why stop then? Continue intimidating, continue bargaining. Roll again. They could further succeed in getting Sildar back for free, or maybe they would roll badly now and fail. Point being: if they are unsatisfied with a result, they can try to improve that result in game, not complaining about it to the DM and saying it's unfair or whatever, and the DM can guide it so the player (and the character) is motivated to try harder and get even more.
Lost Mine of Phandelver is designed for a group of four or five characters, not three. Keeping the encounters as they are is effectively making the campaign considerably harder than it was designed. The 6 goblins are worth 600 Adjusted Experience against 3 level 1 characters, and that is not even considering the HP buff on Yeemik. A Deadly encounter for 3 level 1 characters is 300 XP. And that is happening after the Goblin Ambush, possibly fighting against some Wolves, and the goblin scouts in the entrance of the cave, and, since you already got the frog jade statue, after defeating Klarg (the Bugbear). Thus, you are completely right that the goblins have a huge advantage in this encounter and would probably not be scared easily by only three puny-looking adventurers.
But I am saying that so your DM/GF can keep that in mind: if you guys are running the adventure with 3 players, she will have to re-balance some encounters, or the campaign will be harder than you may want.
Sometimes, Giving in is okay
I will also add this suggestion as someone who has had many problems similar as yours: Sometimes, giving in to the player expectation and allowing them to have it their way, unless it is going to completely break the game by doing that, is completely okay. In this scenario, allowing the players to get Sildar back for free is not game-breaking, and if it was going to keep the table from arguing nonstop with little reason, honestly, I would have done so. I am not saying the ruling was unfair, and I am certainly not saying the ruling was wrong. I am saying that ruling otherwise might have mitigated many problems that didn't need to exist. The player didn't behave the best way either, but it's not the player who is asking the question, it is you, in behalf of your GF, so that's how she can improve: just give in sometimes. Allow the player to feel rewarded.
It's just 40 gp worth of gold. In the next few days you will be getting waay more than that in the adventure. The players don't know it, so they may actually be worried about the gold, but the DM knows, so they shouldn't be worrying so much about it.
Sometimes, it annoys me, as the DM, to give in to something that may not make complete sense to me (like in this case, why the goblins in numerical advantage are scared enough to give up that easily? I don't know), but honestly, it annoys me way more to have to argue with a player for minutes and get people angry, frustrated or leaving the table because of that.