Why is she named "Drummer"?

^{Note, I haven't seen season 5 yet - saving it up to box-set. However I've read all the books several times.}
I don't think it's ever mentioned, in the series or the books. She was a bit of an incidental character in the books, first appearing in book 5; she didn't get a first name until book 7.
Ref: The Expanse Wiki
In the series, like they did with Avasarala, they brought the character into the timeline much earlier & (probably, though I cannot remember precisely) made her a bit of a compound character, like Anderson Dawes*, to save having a hundred tiny characters you'd never remember.

The Surname Database claims it originated as a place name, not an occupation.

Excerpt:-

… the name is locational from a place called Drummer, near Chadderton, in Lancashire. The meaning of the village name is obscure but it is probably from the pre 7th century Olde English 'drum' meaning a ridge.
… We do not think that the name however spelt was ever occupational or even a nickname for a drummer. That word is derived from the Dutch 'tromme' and was not used in England until the16th century, several centuries after the usual introduction of surnames. However for some nameholders, it is a possibility

*Dawes is barely in the books. For the TV series they gave the excellent Jared Harris pretty much all the story lines and character speeches from a plethora of individually unimportant minor characters, to give the OPA a central voice the audience could more easily relate to, and not incidentally a central Belter accent that everyone else had to try to copy, rather than the myriad accents in the books.

It's just an existing surname.

That's really all there is to it. You're thinking of one possible origin, but it really isn't the only possible one.

Other characters have pretty 'typical' names

As with many realistic shows that follow humanity in their spaceflight stage, people's backgrounds, accents, and heritage (culturally, racially, ...) tends to showcase a mix of all major human "categories" (for lack of a better name).

Drummer may be a Bavarian name, but it's not like international surnames don't exist in English speaking regions either. So while the show may display more international names than is statistically probably for the English-speaking world, that's resolved by understanding that a future human society is likely to have mixed and mingled more than our real world society has today.