Or you could call it the “X phial” :: ducks ::
Anyone you could recommend Stateside who might want to take a look at a long-ish first fantasy novel, what with you being a published author’n’all? 
I wouldn’t call myself a published AUTHOR. The writings I sell tend to be either boring consumer advice pieces or short stories to markets like this.
How about the Dumb Flask, using the archaic definition of dumb as meaning “mute.”
You could also work in a cheap joke or two 
Whatever it is called, it should be made of gold.
Personally, I think it should have a name which cannot be spoken.
No, I want the innards to be visible. Later in the story, when the children finally find the stolen voices, they will be contained in a device called the Engine of Song.
Then it should be made from dichroic glass
Looked at straight on it appears to be a mirror, turned one way, its blue, turned another its pink. Held just right its clear as… well, glass.
On the website much of what they show is an opaque style, but it does come in clear. Its very expensive (read precious)
I still like Saturnine phial, it sounds slightly sinster, yet haute and distant
Make it look like a tiny harp, made of metal.
When the strings are plucked, one or more words, sung by the character’s voice, are heard.
Call it Minstel’s Ruin.
A good idea, though not for this story. I will, ahem, borrow that concept for another one, though, if I may.
Explain the Saturnine part to me, if you would.
Similar to taciturn, but colder, slower. Sardonic, gloomy or surly. It implies smug superiority to me, but that’s just me.
How about the Aural Vial? You could make it gold, with the Au being the symbol for gold, which could be a clue. Although Aural really pertains more to hearing than to speech - How much more frustrating would it be that she could speak, but what has been stolen is everyone’s ability to hear her words. She knows she can be heard, the animals hear her, just not other people.
Whoops - I didn’t mean to rewrite your story.
StG
An aphonicon?
Oh, I don’t mind, as I freely ignore any idea you don’t like and use ones I do. It won’t work here, though, as the character in question, though pivotal, is not a viewpoint character. Before this point in the story she has been the caretaker and defender of her younger brother and his best friend, who must afterwards go on a quest to restore her voice and save her life.
And with that I’ll stop talking. er, typing.
If it’s an Engine of Song, how bout something musical? Sotto Vocce?
Flask of Laryngitus?
Mumble Jug
I don’t know if it’s right for the story but it’s fun to say.
Ommalimluch, the stealer of breath
Or it could conceivably translate as the Breath Djinn, stealing voice for dire ransom in exchange for release from the bottle. His escape a scourge upon the earth. Could be an opportunity for his clever containment within the story. Outwit the Djinn and steal back her voice with some turn of a phrase.