Yes, using Tides of Chaos is legitimate in this situation, at least until we get official errata that states otherwise (I could not find any). While you generally can't do "anything" whilst at zero hit points, you're still clinging on to (what may be your last moments of) life; you can still use your wild magic to tip the scales in your favor.

Tides of Chaos is not an ability, spell, action, or reaction, but instead is a class feature, like a rogue's Sneak Attack, for example (i.e. it modifies a specific circumstance in the class' favor). Since the RAW states it can be used for saves, this surely must include a Death Saving Throw, because it is classified as a type of save.

Granted, you can realistically only use it one time until you gain at least one hit point, since a recharge is only allowed after casting spells or a long rest, but I think it'd be exciting to use it right when you have two failures and two successes, barely pulling your character back from the brink of death...

The gist: your Armor Class remains the same despite being unconscious. You can justify a miss by saying the unconscious target was not hurt by the attacker's careless attack.

Yes, your AC is still kept the same

Your thief's AC is 15, before and after succumbing to the sleep spell. That is simply how AC is calculated (Base AC of Armor + Dex mod), as written in PHB 144.

PHB 144

Armor Class (AC). Armor protects its wearer from attacks. The armor (and shield) you wear determines your base Armor Class.

Followed by the AC specified in PHB 145 on how to calculate Armor Class from Padded or Leather armor: \$11 + \text{Dex modifier}\$

How is this narratively justified?

Armor Class is not how hard you are to hit, but how hard you are to wound.

Armor Class, PHB 14

Your Armor Class (AC) represents how well your character avoids being wounded in battle.

The difference is usually immaterial, but in this case, it is important. The Barbarian could still have hit the Thief, they just failed to hurt them.

We can also take guidance from how Armor Class is calculated for objects:

Statistics for Objects, DMG 246

Armor Class. An object's Armor Class is a measure of how difficult it is to deal damage to the object when striking it (because the object has no chance of dodging out of the way).

Objects cannot dodge, much like a sleeping creature, so you instead use AC to represent how hard it is to damage the object. Interestingly, the following ACs are used for the brittle materials:

\begin{array}{|c|c|} \hline \textbf{Substance} & \textbf{AC} \\ \hline \text{Cloth, paper, rope} & 11\\ \hline \text{Crystal, glass, ice} & 13\\ \hline \text{Wood, bone} & 15\\ \hline \end{array}

Your unconscious Thief, as it turns out, has the same AC as bone. This means, even after being hit by something (because it can't dodge), bone is still difficult to damage. It is probably resilient enough that you cannot crack or snap it in one blow.

In other words: you can let your Barbarian's swing hit the Thief, but you can justify by saying, despite the impact, did not actually hurt the target. They could bruise, or get a cut, or a small puncture wound, but the nature of the blow is such that the target was not meaningfully harmed.

Also narrate the failure of the Barbarian

Despite having Advantage, your Barbarian's best roll was still a 14. That is a failure on their end, not just a product of the resilience of your unconscious Thief. Focus on that as well.

You can say that the Barbarian thought they were aiming for the head, but they lost their footing slightly during the swing, or aimed too quickly or too excitedly, or the hilt on their weapon slacked just a bit. You can also add an external factor to the blow, and use that external factor to explain why they failed to deal damage.