The title is the question.
I figured it was Irish, so I threw it into Google Translate – for those who don’t know the term (like me), it translates to “bold boy.” But, Google Translate doesn’t do Irish pronunciation.
This site has several native Irish speakers pronouncing “buachaill” – depending on the speaker, it sounds kind of like “buh-hoy” (more often) or “buh-hull”, with maybe a bit of a “ch” sound at the start of the second syllable.
I’m no expert on Irish pronunciation, but if no one else is able to help, I do have a couple of Irish friends who have varying levels of mastery of the language, and I can ask them tomorrow.
At first glance —
[bʷɑxəlʲ d̪ʷɑ:n̪ʷə]
You’ll not get very close unless you know something about Gaelic languages. But to English speaker it might sound like
baw-ch’ll daw-nuh
In which the “ch” indicates a sound as in Scottish loch or German ach
Interesting that the labialization of the consonant seems to more affect the vowel.
Exactly as it’s spelled. 
You’ve got some excellent guidance from previous posters but I would suggest boo-ch’ll dhaw-nuh would approximate the first vowel more closely.
Your observation about how the broad/slender consonant distinction (not labialisation) is perceived as a difference in vowel quality is very interesting.
I’m not entirely sure what you mean.
First, the consonants aren’t labialised. In Irish, every consonant is paired, with a palatalised (“slender” or Y-colored) and a velarised (“broad” or W-colored) version.
Second, I’m not seeing where you see it changing the vowel. The basic vowel quality doesn’t change except that sometimes palatalisation inserts a [j] glide before the vowel.
Unfortunately, I can’t read that.
Interesting. Those are what I would have guessed. Thanks.
Perceiving the broad/slender consonant as a change in vowel quality is not unique to BigT. It seems to be very common among non-native speakers (i.e. almost everyone). Some common examples you might hear on the news in Ireland:
Dáil (the lower house of the Irish parliament) pronounced as “Doyl”
Fine Gael (an Irish political party) pronounced as Fi-nye Gwayle
It’s somewhat similar to non-native speakers of Spanish hearing (and pronouncing) the word año as “anyo”.
I said labialization but meant velarization. (I get those mixed up because, to me, [w] is primarily about the lips.) And I interpreted your use of “aw” as a different vowel than the [ɑ] as written (which I would tend to write as “ah”).
I would usually write “ah” as [a]. (“True ‘ah’” I would transcribe more precisely as [ä] or [ɑ̈], with a preference for the former to distinguish it from [ɑ]) This transcription is not an exact science. The Irish [ɑ] can tend toward [ɒ], but it’s not fully rounded, so it is easily perceived by English speakers as an “aw.”
Velarization is more correctly transcribed as /bˠ/ but /bʷ/ was as close as I could get on the tools I had available to me when I wrote that post.
As someone who spent 14 years learning irish in school this is the closest out of the attempts so far
Always try forvo.com for these things.
Here’s six different Irish folks pronouncing buachaill:
And the top one is dána in Irish:
Throatwobbler mangrove.