Theory: We know (most of) the names Kvothe knows already.
When a True Name is written in the books, it's done in italics. Every time Kvothe calls the Name of the Wind in the series, we see "Aerlevsedi". This is a Name we know he knows for certain, so we'll start there. From NotW, ch 84:
- He looked at me. His dark eyes steadied me somewhat. Slowed the storm inside me. “Aerlevsedi,” he said. “Say it.”
- “What?” Simmon said somewhere in the distant background. “Wind?”
- “Aerlevsedi,” Elodin repeated patiently, his dark eyes intent upon my face.
- “Aerlevsedi,” I said numbly.
Kvothe knows the Name of Wind, so we see Aerlevsedi but Simmon doesn't - so he only gets "Wind" (no italics). For comparison, when Elxa Dal tells Kvothe he knows the Name of Fire - something Kvothe can't conceive of understanding, as he ends up going into the chemical properties of what makes up fire with Dal when trying to understand how he can know the name of Fire - we instead see this (WMF, ch 22):
- Dal hesitated for a moment, then smiled. He looked intently into the brazier between us, closed his eyes, then gestured to the unlit brazier across the room. “Fire.” He spoke the word like a commandment and the distant brazier roared up in a pillar of flame.
- “Fire?” I said, puzzled. “That’s it? The name of fire is ‘fire’?”
- Elxa Dal smiled and shook his head. “That’s not what I actually said. Some part of you just filled in a familiar word.”
Note the lack of italics here. PR tells us, through Dal's conversation with Kvothe, that if it's a Name we don't know, a familiar word is filled in instead. Remember the previous quote. Kvothe said the Name of Wind, and although didn't yet understand it, we still saw Aerlevsedi. Fire, on the other hand, is a name he doesn't know, so we only see Fire - no italics.
Kvothe is the one telling this story, as Kote. So what we know, he knows. Chronicler is likely writing "Wind" and "Stone" and "Fire" as he writes it down, since that's what he'd hear - but Kote is the one telling it. Kote knows the Names, so that is why we see them when we read the story. It is "an accurate, truthful account" of Kvothe's life, and so Kote is going to use the real Names.
- Chronicler’s face was calm as he pressed the metal disk firmly onto the table with two fingers.
- “Iron,” he said. His voice sounding with strange resonance, as if it were an order to be obeyed.
- Bast doubled over as if punched in the stomach...
Kote/Kvothe doesn't know the Name of Iron. And, as the frame story is happening in "real time", all we see is "Iron" - no italics. This is confirming, yet again, that if Kvothe does not know a Name, we only see what the mind fills in as an appropriate substitue.
In understanding all of this, we can use the presence of italics in the books to determine all the Names Kvothe has learned through D3 that have come up in the story so far. In WMF, chapters 83, 93 and 151, Kvothe knows the Name of Roah (explanation below) wood.
In ch 83 of WMF:
- So Taborlin went over to the chest, but it was locked. Then Scyphus laughed and so did a few of the guards.
- “That made Taborlin angry. And before any of them could do anything he struck the top of the chest with his hand and shouted, ‘Edro!’ The chest sprung open and he grabbed his cloak of no particular color, wrapping it around himself.
The above quote is when Marten is telling the story around the campfire - but don't forget that it's a story within a story. Kote is relaying the telling of this story to Chronicler, and he knows the Name of the wood used for these chests, so we see edro. In ch 93 of WMF, we have:
- I thought back to the story Marten had told days before. “Of course,” I laughed, then shouted, “Edro!” in my best Taborlin the Great voice and struck the top of the box with my hand.
- The lid sprung open.
In this case, it can be simply an example of Dedan and Hespe fiddling with the lock and that's why it opened. However, the chest can also be made of Roah (or a similar composition hardwood) since it is described as "a heavy hardwood box". It doesn't matter whether or not he used Naming to open the Maer's money chest - what does matter is that Kvothe used the Name of the wood when telling the story to Chronicler. It confirms, again, that he knows this Name.
In WMF, ch 151:
- After replacing the keys in their box, he put his hands back on the sides of the lid in the same position as before. “Open,” he said under his breath. “Open, damn you. Edro.”
- He lifted, his back and shoulders tensing with the effort of it.
I say that the Name of the wood he knows is Roah is because his chest in the Waystone is Roah wood. (It might simply be the name of Wood that he knows.) While he may no longer have the power to command it as Kote, he still remembers the Name from when he was Kvothe. Remember back to Marten's story about Taborlin the Great. Kvothe doesn't know the name of the stone used to build walls (we'll say Cement, for argument's sake), nor the name of the wood the doors were made of (again, for ease of understanding, we'll guess it's Oak - or maybe the original story used the Name of Fire, another that Kvothe doesn't know) because when he tells the story of Marten telling this story to Chronicler, we only see "burn" and "break" - yet we still see Edro for opening the chest Scyphus has Taborlin's tools in:
- But Taborlin the Great knew the names of all things, so all things were his to command. He said to the stone: ‘break!’ and the stone broke. The wall tore like a piece of paper, and through that hole Taborlin could see the sky and breathe the sweet spring air.
- “Taborlin made his way out of the caves, into the castle, and finally to the doors of the royal hall itself. The doors were barred against him, so he said, ‘burn!’ and they burst into flame and were soon nothing more than fine grey ash.
- ...he struck the top of the chest with his hand and shouted, ‘Edro!’ The chest sprung open...
The only italicized Name in the above quote is the one for the wood the chest is made of. It confirms that Kvothe does not know either the name of the wood used for the doors (or, from talking with Elxa Dal, that Kvothe does not know the Name of Fire), nor the name of the type of stone (likely cement of some fashion) that walls are made of.
However, in NotW, ch 26, Kvothe knows the Names of Stone, (the natural element, not the man-made one used for walls), Air (different than Wind), and that of Selitos himself:
- “Silanxi, I bind you. By the name of stone, be still as stone. Aeruh, I command the air. Lay leaden on your tongue. Selitos, I name you. May all your powers fail you but your sight.”
It is not impossible that Kvothe can know the Names of people. He's able to Sing the name of Felurian, after all, and in NotW, ch 16, we see that Kvothe also knows the Name of Cinder:
- The soft voice went as hard as a rod of Ramston steel. “Ferula.”
- Cinder’s quicksilver grace disappeared. He staggered, his body suddenly rigid with pain.
And he also knows the name of the Moon, Ludis, from WMF ch 88:
- She leaned close and spoke warmly against his ear, “Ludis.”
- And Jax brought out the black iron box, closing the lid and catching her name inside.
In NotW, ch 46, while Kvothe doesn't know the Name of the Stone used to build walls (here he sees "BREAK" instead of a Name), he *does* know what the stone becomes when it's been corrupted by Copper. He hears "BREAK!" the first time when Elodin tries using it on the type of stone in Haven (a Name he doesn't know, like the cement example above) but when Elodin actually breaks the wall, we see something else entirely - a Name Kvothe does know:
- “But Elodin the Great knew the names of all things, and so all things were his to command.” He faced the grey wall beside the windows. “He said to the stone: ‘BREAK’ and the…”
- Elodin trailed off, his head tilting to one side curiously. His eyes narrowed. “Sod me, they changed it,” he said quietly to himself. “Huh.” He stepped closer to the wall and lay a hand on it.
- “Oh,” Elodin said suddenly, laughing. “That was half-clever of them.” He took two steps back from the wall. “CYAERBASALIEN.”
Three other points of interest are related to this. Kvothe knows the Name of Stone, as it is when combined with Copper, shown in the above example with Elodin. What else has he studied as often as the Wind that is Stone+Copper? ...the Valaritas door in the Archives. Assuming only Naming is needed to open it, Kvothe has shown he already knows the Names he needs to do so.
And, we know from watching Kvothe learn the Name of Wind that Kvothe learns Names by studying them. Why would Kvothe have reason to study the walls in Haven later to learn the Name of Stone+Copper... unless he became a resident of Haven himself? What if the direction Kvothe's life takes in D3 where "things get darker again" is his time in Haven?
Lastly, we know from TSRoST that Auri can read the future (evidenced by her knowing Kvothe would come a day early from the way the gear broke, and proven by the music we hear at the end of the story), and she's prepared a place for Kvothe in the Underthing - knowing he will need it at some point in the future. The Black Door that she's so afraid of? Probably leads to Haven. Haven, where she wouldn't go for herself, but would for Kvothe.
So, in summarization (TL;DR), Kvothe does not know the Names of Fire or Iron, but does know the Names of Wind, Air, Stone+Copper, (Roah?) Wood, Cinder, and Selitos.
- Aerlevsedi = Wind
- Aeruh = Air
- Silanxi = Stone (the natural element)
- Cyaerbasalien = Stone+Copper
- Edro = Hardwood (Roah, probably also similar composition hardwoods)