It’s in the Scot-Irish tradition, and they’re easier to build than fiddles, so why did not not make it across the Atlantic along with the other instruments in that musical tradition?
Why the fuck* is this in the BBQ pit?
*Well, it is in the pit.
I already reported for forum change.
And for a more serious answer, bluegrass doesn’t really need that sort of rhythm instrument, as it’s not dance music. No idea why it didn’t appear in earlier American dance music (stringband stuff, and the like).
Is the bodhran really Scots-Irish? Aren’t the drums of the Orangemen somewhat larger?
The fiddle came from the British Isles to North America. The banjo came from Africa. Early string bands & minstrel groups (black & white) used both instruments & added the guitar & mandolin. These styles evolved into Bluegress–thanks Mr Monroe! The slide guitar came from Hawaii & is also found in Bluegrass. (Electrified as the pedal steel, it accompanied the fusion music known as Western Swing. Then into Honky Tonk & Mainstream Country–although Modern Nashville considers it a bit old fashioned.)
The bodhran has ancient roots but really became popular in the Irish music “revived” by Sean O Riada–where the Chieftains got their start. One might as well ask why Bluegrass bands don’t include uilliean pipes & pennywhistles. Of course, the Chieftains have recorded with folks who play all the above instruments–& more!
It’s an interesting question, and the correct and useless answer is “because Bill Monroe didn’t have one”. Monroe’s lineup of mandolin/guitar/fiddle/banjo/bass is considered definitive when it comes to bluegrass.
A better question is why you don’t see the bodhran in old time music, which is the immediate precursor to bluegrass and which remains alive and distinct in my part of the world. You do occasionally see traditional jug band instruments like the jug, bones, and spoons played in old time. Monroe originally had someone in the Blue Grass Boys who played those instruments, but it didn’t last long; he probably thought it interfered with the mandolin “chop” that’s so characteristic of the bluegrass sound he was creating. And you don’t see those instruments in old time jams much anymore, which is probably an influence that bluegrass has had back on the original genre.
Also, the bodhran is hardly the only traditional Celtic instrument that didn’t make it to bluegrass/old time. I haven’t seen many bluegrass bands with a tin whistle, either.
Any Lambeg drums in Bluegrass? 
Moved to CS.
In general (not that there weren’t exceptions) country music didn’t use drums and, in fact, they were at one time banned from the Grand Ole Opry.
And southern blues evolved without drums because slaves weren’t allowed to use them.
In fact, it’s better to phrase the question as “why are there no drums in southern music?” And to thank Bob Wills for his under-recognized contribution to percussion.
The Wikipedia page confusingly states both that the bodhran is a new instrument, developed in the mid-20th Century (which would pretty much answer the OP), and also that the word was first used in the 17th Century. 
The slide guitar didn’t really come from Hawaii. It’s use in the “traditional” Hawaiian sound Is a relatively late innovation. In fact, the guitar in any form only arrived in Hawaii in the 1800’s, brought by Mexican cowboys. the slide/string concept originated in Africa, per Wikipedia. Native Hawaiian music apparently used percussion and voice.
In bluegrass and old time string music, the bass tends to take over the percussion duties, along with, at times, jugs, mouth harp and spoons. In some of the gospel string bands you’ll also see the tambourine. Hand claps and finger snaps are used as well.
As kunilou pointed out, most southern music up until Bob Wills’ western swing didn’t use drums, including not only string band, bluegrass and blues, but traditional Cajun as well.
I never heard of a bodhran before.
Learned a new word today thanks to the SDMB.
Why is the OP suggesting a drum would **replace **a violin? A violin plays melody lines. A drum is a percussion instrument.
No; the Scots-Irish only brought their fiddles.
But the bodhran reflects the self-consciously “Celtic” strain in Irish music. Not the Scots-irish one…
I think that quote isn’t saying it was invented in the 1960’s, just that previous to that it would have thought of it more as a noisemaker or toy (like a kazoo) rather than an actual “legitimate” musical instrument.
And it’s not easy to hold a drum under your chin.
Bluegrass has been tied to dancing for as long as it has been around. Buck dancing and clogging are performed almost exclusively to bluegrass music and square dancing has it’s share as well. As noted by missred, the beat is kept by other instruments.
I may be using a stricter definition of bluegrass than you, but all those forms of dancing greatly predate bluegrass, and to my (rather limited) knowledge wouldn’t even be danceable to it. The usual tempo of bluegrass, apart from anything else, would make any sort of extended formal dancing to it impossible.
I’ve spent many a night in a bluegrass bar in the late 1970s and watched people of all walks of life dance/clog quite successfully to bluegrass. Bluegrass tempos can be varied enough to accommodate clogging specifically… Buck dancing may be a bit more problematical.
It appears that the word has been around longer than the instrument that currently goes by that name. Which would make sense if it originally was applied to a variety of (sometimes improvised) percussion instruments, and was later adopted for a newly-invented (or possibly just standardised) instrument of that type.
I’m just going by the same page you are, though, so who knows – as you say, it’s not clear.