What does "Boh!" mean, precisely?

"Boh" means quite simply "I don't have a clue".

BUT

often saying it implies "Why the heck are you asking me, since there is no reason whatsoever for which I am supposed to know it"

So, there might be a light intent of reprimand towards the asker when someone says "boh".

I think that the closest translation in English should be:

dunno

So in formal English is translated as "I don't know".If you analyze some example of dunno form merriam-webster they can be translated with boh :

“Whatʼre those things?” whispered Ron. “Dunno,” said Harry. — J. K. Rowling, Harry Potter and the Order of the Phoenix, 2003

Cosa sono quelle cose? bisbigliò Ron. "Boh..." disse Harry

From the wiktionary

Puoi pagare il mio biglietto? Boh, devo vedere se ho i soldi.

Can you pay for my ticket? I don't know, I'll need to check if I have the money.

Any other meaning is purely a conjecture

It's not implicit any other meaning like "I know but I am not gonna tell you", "I don't know and I don't care", "I am confused". So if Italians say to you "Boh" usually they not trying to hide you something or avoid answering, they just don't know.

I live in Rome. The word "boh" usually is used to say "I don't know". But you have to pay attention in which contest is used. In fact it could be used to say: "I have no idea". This last mean's different from the first cause it is more general.

-- What time does the match start?
-- Boh (I didn't know it will be a match!)

I hope you will understand! :) Good bye

I was watching an Italian movie once, in Italian, with Italian subtitles. If I remember the name of the movie I'll edit this post later. One of the characters said "Boh!" in conversation, and for some reason the subtitles translated it to "Non lo so!"

I feel like the two best translations to English (depending on context) are "Meh" and the humming variant of "I dunno," although I have no idea how to spell that. If you're the speaker you can use it for either/or.

Boh is used to mean uncertainty, indifference, disbelief, contempt; it is similar to bah which is used to express uncertainty, hopelessness, disbelief, contempt, and mah that express doubt, uncertainty.

The first two words are onomatopoeic, while mah comes from Latin măgis.

Boh means doubt, indifference, unwillingness to talk about something or to express an opinion. It's widely used in the Roman dialect (but not only). One possible explanation of its etymology is the sound that one produces when wants to express uncertainty.
A non-native speaker should pay attention to the context, where Boh! appears, and then it's quite easy to deduce its meaning, because it usually appears in response to something.

'Boh' means 'I do not know', with all its nuances:

• I don't know,
• I know but I am not gonna tell you,
• I don't know and I don't care,
• I am confused
• etc.

I've been looking around for the etymology of this word but it seems not to be available.