The “prickmouse” and the “butcher's broom”

Solution 1:

An important point relevant to this question is that the earliest meaning of the word "broom" in English, dating to c1000, referred to a particular species of flower. It appears that "broom" as a botanical term in English was not a metaphorical extension of tools for sweeping, but rather the original meaning, figuratively extended to refer to the tool.

Per the OED:

A shrub, Sarothamnus (or Cytisus) Scoparius (family Leguminosæ), bearing large handsome yellow papilionaceous flowers; abundant on sandy banks, pastures, and heaths in Britain, and diffused over Western Europe. Also the genus to which this belongs, and the allied genus Genista, including the White Broom, and Giant or Irish Broom cultivated in gardens, and many other species.

The OED proceeds to offer an extended definition that includes more plants as definition 2:

Entering into the name of various other plants used for sweeping, or in other respects fancied to be akin to the broom proper; as butcher's broom n., Spanish broom n. (a kind of grass).

Note that in this context, "broom proper" likely refers to the plant Sarothamnus Scoparius, not "a utensil for sweeping things," which the word would come to mean later, first recorded in the 1400s.

Far more botanical brooms are discussed in modern texts, including in Wikipedia.

Because of this evolution of the word "broom," the OED makes only a passing reference to the specious etymology provided by The Rural Cyclopedia, edited by John Marius Wilson, which contains this excerpt on "butcher's broom."

The whole bush is gathered by butchers, and made into besoms for sweeping their blocks and shops; and hence it obtained the name of butcher's broom.

The complete text from the journal can be viewed here on Google Books.

It seems notable that this citation is from 1847 whereas the term "butcher's broom" dates back to 1538 (and, as noted above, broom itself dates back to c1000), so the veracity of the Cycolpedia's etymology shouldn't be assumed infallible.

Prior to offering the 1847 citation as comparative evidence, the OED makes its more primary etymological assertion:

Apparently so called on account of the appearance and texture of the stiff, spiny tips of its leaves

Long story short?

That long-handled implement that witches and wizards play Quidditch on was likely named after a kind of flower, not the other way around.