This is a remix of an older answer:

Let's assume that by 'creative names' you mean a 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), or associated with the area or venue where that group most often plays (e.g., The Savoy Havana Band of the 1920s, so named because they were the house band at the Savoy and played music associated with Havana, or the Original Dixieland Jass Band of the 1910s, who, well, claimed to be exactly what it said on the tin). One that's abstract, in other words - Buddy Holly's band The Crickets, as you mention, aren't actually insects.

In this sense, the band name has been around in recorded pop music since 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 naming tradition came about, exactly, but I suspect it goes back to the big band tradition (or elsewhere in 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.

But broadly speaking, in terms of popular music, the overwhelming majority of music in the pre-rock era was released under the name of Individual Person or Individual Person And Their Orchestra; the likes of The Ink Spots weren't the name. This changed, 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. And The Crickets did continue after Buddy Holly passed away, releasing successful singles like 'Please Don't Ever Change' in the early 1960s - at that point they were using an abstract name without naming their lead man.

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.

By around the mid-1960s you get groups who not only have abstract names but who drop the The at the start and the 's' at the end - bands that initially were labelled 'The Cream' or 'The Pink Floyd' start to become known as 'Cream' and 'Pink Floyd', and you get, for example, a group like Love releasing music in 1965. The Beach Boys, at one stage in the late 1960s/early 1970s, considered renaming themselves Beach. I mean, this whole topic is literally all semantics, and you could quibble with the semantics, but there's something even more abstract about 'Cream' than 'The Beatles' - where 'The Beatles' tells you that it is a group of musicians, names like 'Aphex Twin' or 'Cream' tell you even less about whether the music is made by a solo act or a group.