Been meaning to quote this one for a while:
欧: Ō: Europe, Eu-- Formerly 歐. 欠 is lack here in its literal sense of gaping mouth. 區/区 is ward/section. Though the 品 of 區 probably referred initially to various enclosures, because of the similarity to mouth 口, 區 was often chosen as a phonetic in words relating to the mouth.Ewww-rup, obviously. (And if you look in the wordtank, they give you the reading 'ha(ku)/ to vomit' right next to the ō of ōbei the West.)
Here it is used to express the sound Ō. (Though as an independent character, 區 is now invariably read KU, it is also listed as having the minor reading Ō, both readings appearing to stem from an original reading of YOKU/ EOKU/ EUKU or similar.) Thus to make the sound Ō (EO/ EU) with a gaping mouth, a reference to groaning while vomiting. (欧) can still be used to mean vomit in Chinese, while in Japanese this is expressed by the character 嘔, which uses an ordinary mouth 口 instead of a gaping mouth 欠.
The character was also chosen as the phonetic for the EU of Europe, as well as Eustachian, Euclid, etc. Why a character of such emetic connotations should be chosen to represent Europe is a matter of some conjecture. While it is true there are very few characters with a reading of this particular type of Ō (EU/EO, as opposed to OO, OU etc.) it should be noted there is a perfectly good non-general use character 謳 (言 is words) which is read Ō/EU/EO and has a meaning of praise or extol.