This question has been asked a few times before, the best answer I thought was this one talking about the Richardson Rock Band circa 1842.
There are links to a few other similar questions in that post, but nobody else has anything prior to early 20th c.
Great question - I hope someone manages to dig up some more info on this topic!
I tried to answer a similar question here a year ago, and so this comment is a slightly altered version of that to reflect the different question (and a little further research on my behalf).
If by 'specific name' you mean an abstract name that isn't directly related to the band members (e.g., that Maroon 5 are called Maroon 5 and not Adam Levine And His Orchestra or Adam Levine And The Five Maroons), the band name has been around in recorded pop music since the at least the 1930s as a way to name vocal groups like barbershop quartets. The Ink Spots, for example, formed in 1934, and became prominent after 1936 (according to this page). This tradition of naming vocal groups with specific abstract names continued into the fifties/the doo-wop era, when the likes of The Penguins (e.g., 'Earth Angel') and The Chords had hits, and it was this tradition that extended into, say, the Motown tradition of giving vocal groups names like The Marvelettes or The Temptations or The Supremes.
There's also a parallel tradition of instrumental rock bands in the late 1950s and early 1960s having abstract specific names - there's The Shadows, The Fireballs, etc. I'm not sure why that instrumental rock band tradition came about, but presumably there's a lineage from those bands to jazz. There doesn't seem to be much of a tradition of jazz bands with specific bands beyond Someone And The Somethings or Someone And His Orchestra - there are the likes of The Modern Jazz Quartet and The Quintet, but it doesn't really become a trend in jazz in the same way it does for rock music, and 'Modern Jazz Quartet' is descriptive in a way that 'The Penguins' is not.
In terms of pop/rock music, The Beatles are ground zero for the popularity of the 'band name' a la Maroon 5, simply because of how enormously successful and thus imitated they were. The Beatles have talked in interviews about how much they loved the idea of The Crickets (this is discussed reasonably extensively in the quite scholarly Beatles history Tune In by Mark Lewisohn), who were Buddy Holly's backing band. Because Buddy Holly & The Crickets' music was being released on two separate record labels, their releases ended up being released under the name 'Buddy Holly' on Coral Records, and 'The Crickets' on Brunswick Records. However, the Beatles in England wouldn't have known the circumstances of why The Crickets were different to Buddy Holly, and so they saw The Crickets as being an idea of a band that they wanted to follow.
Also very influential on the idea of the band name was The Beach Boys, whose first single in November 1961 predates the recording career of The Beatles - The Beach Boys had national hits before The Beatles broke through in the US. The Beach Boys were an odd combination of doo-wop vocal group and a surf instrumental band, both of which had traditions of abstract band names (see above). The name 'The Beach Boys' was chosen for them by someone involved in the local record label, Candix, that their first single was released on, but they had previously called themselves The Pendletones.
Between the success of the Beach Boys and the Beatles in 1962-1964, record companies fell over themselves to sign up acts with such band names. One way to see the change in this period is to compare the UK charts at the end of 1961 and at the end of 1965. The UK singles chart at the end of 1961 is predominantly solo artists, especially towards the top. There are some band names - most prominently the instrumental rock group The Shadows, but solo artists dominate. By the end of 1965, there are seven bands in the top 20 of the the UK singles chart, from the Beatles to The Who to The Four Seasons to The Toys.
While many performing ensembles have always had definite names, it seems to be the case that historically, many musicians were either connected with a certain performing establishment (The Vienna Opera Orchestra) or liturgical group (St. Mark's Cathedral Choir).
After Haydn brought the string quartet to prominence, with his work with the court musicians on hand when he was composing for Baron Carl von Joseph Edler von Fürnberg sometime around 1755-1757 chamber music flourished well into the 19th century and I'm sure that many performing ensembles had "band names" especially if they were prominent string quartets, whose composition, interestingly enough, had fallen out of favor after Beethoven died, but that is merely an interesting side-note.
Oftentimes in the 19th century the virtuoso would be the main focus of the performance (Guys like Pianist Franz Liszt and violinist Niccolò Paganini) with a backing orchestra or what have you.
In the Jazz age the orchestras would be named after their bandleaders most of the time, Count Basie and His Orchestra, and then when Bop became popular it was all about the lead soloist again, Charlie Parker plays with the gang, or something of that nature.
Because Jazz musicians often performed with a variety of different people, this made sense from a marketing perspective. Keep in mind, that these musicians were performing a common repertoire, as opposed to original songs arranged by the group (ala The Kinks).
The Skiffle Bands and early rock and roll bands continued with the bandleader format such as Little Richard and Elvis Presley, Johnny and the Hurricanes, etc.
The Rock and Roll tradition is the first time where we see bands naming conventions breaking away from this, with the Beatles.
I do not have an extensive knowledge of who did it first, necessarily, but as you know, that the popularity of the Beatles contributed heavily to the staying power of this naming convention moving out of the 1960s.
Are you asking about a group of musicians known by a collective name, like The Beatles or The Who, or a group of musicians associated with a particular person, like the Muddy Waters Band or Glen Miller Orchestra?
If we're looking at pop music groups, we must start with the roots: Jazz combos and big bands, country & western acts and finally rock n' roll bands.
We see the earliest roots of a "band brand" in Jazz. In 1926, we have "Jelly Roll Morton playing with - not the band with Flea - "The Red Hot Peppers". By early 1930s, there's "Louis Armstrong and Hot Five" and "Andy Kirk and the Twelve Clouds of Joy". (By the early 1950s, you have Sun Ra and his Arkestra, the band's name clearly signaling the unconventional musical styling. This is perhaps the beginning of the convergence of musical style and identity - for instance, it would have been uncool to have an avant garde jazz band named like an old-fashioned combo.)
Even with Country acts, novelty names start to appear in the 1930s: "Hank Williams and the Drifting Cowboys", "The Girls of the Golden West" (one of the pioneering female country duos) and so on.
So this idea of a brand seems to have taken root fairly early, but it undoubtedly became an established marketing practice by the 1950s with the rise of rock n' roll (and perhaps with the explosion of recorded music).
A little analysis of Billboard year-end top singles (Hot 100 for 1959 to the present, and smaller numbers for years prior to then) hits over a 30 year period:
Of the total list of songs, this is the count of how many were by a true band name and not a person or persons. A person or persons would be their real name ("Elvis Presley"), a name and his bandmates ("Johnny and the Hurricanes"), a nickname ("Fats Domino" or "The Blue Barron"), or a group of names ("The Everly Brothers" or "Peter, Paul and Mary").
From the above, you can see that band names started being popularized in the late 1950s, and then became very popular by the late 1960s.
This data used to be on the Billboard web site, but they have since removed data for years prior to 2006.
First question on this is 'what counts as a name?' In addition to what else has been said elsewhere in this thread and in previous iterations, there are certainly orchestral groups from the early 19th or before named after buildings or as 'societies' (Gewandhaus [Cloth House] Orchestra, The Halle, Handel and Haydn Society). In less highbrow circles, the Minstrel shows (c. 1840s) had names which were descriptive of their (supposed) origin but also allowed for differentiation between groups (Ethiopian Serenaders, Virginia Minstrels, Christy's Minstrels, etc.). Similar (reasonably-descriptive) naming was the norm for a lot of late 19thC/early20th C bands we know of (Fisk Jubilee Singers, Haydn Quartet, Excelsior Brass Band, etc.). In terms of names which are not so purely descriptive, there was definitely a boom around the 1920s with Jazz bands (see those in this link here ), and also 1920s 'old time' string bands (Georgia Yellow Hammers, Skillet Lickers, Fruit Jar Guzzlers, etc.). There's probably a further development somewhere down the line where a band name became singular rather than the plural or descriptive of a group (Love, Cream, King Crimson), or even abstract/agramattical (The Who, The The, !!!, etc.).
Second question is 'who is to know?'. Ultimately we know of the orchestras and minstrel shows because their names were recorded. We know of early Jazz and Old Time music because band names were recorded (perhaps made up) to put on newly recorded music. There is not a significant amount of recorded history before these to know what bands were called prior.