"Does the term "maiden name" come from the myth surrounding Kore being abducted by Hades? Because Kore becomes Persephone, so I guess you can say Kore is her maiden name? Is anything more said about this change of name?"

I love the theory, but no, it doesn't as far as I am aware. A maiden name is the last name used by a woman before changing her name upon marriage. Some people prefer to use the term 'birth name'.  As a general rule, a woman's maiden name is her father's last name, reflecting the patrilineal system of descent which dominates many cultures.

The ancient Hellenes didn't make use of surnames, but during some periods, formal identification commonly included place of origin. At other times clan names and patronymics ('son of') were also common. In none of these cases, though, were these names considered essential parts of the person's name, nor were they explicitly inherited in the manner which is common in many cultures today. So, while given names have been used from the most distant times to identify individuals, the advent of surnames is a relatively recent phenomenon. In Britain, hereditary surnames were adopted in the 13th and 14th centuries, initially by the aristocracy but eventually by everyone. In Japan, family names were uncommon except among the aristocracy until the 19th century.

The term 'maiden' has its roots in the Old English. It comes from 'mægden', or 'mæden', meaning a (virgin) girl, a maid, or servant, and/or Proto-Germanic 'magadinom', meaning 'young womanhood', or 'sexually inexperienced female'. The first recorded use of the term 'maiden name' comes from the 1680's, and it was applied to the myth of Persephone long after the myth was first recorded in Greek.

'Korê' (Κορη), or 'Kourê' (Κουρη), is a title (or epithet), literally meaning 'maiden' or 'girl' that was applied to Persephone as a Goddess of spring's beauty. Once She married Hades--kidnapped or not--She became the infernal Queen of the Underworld. An epithet is an attachment to the name of a God or Goddess, used to indicate either a specific domain of the Deity, a specific origin myth or region from which the Deity came, or an entirely different entity, through either domain or origin.

'Korê' remained in use throughout ancient Hellenic history, especially connected to the Eleusinian Mysteries and some of the Eleusinian Festivals. Other cult titles included Khthonia (Χθονια, Of the Earth), Karpophoros (Καρποφοροσ, Bringer of Fruit), Sôteira (Σωτειρα, Saviour), and Praxidikê (Πραξιδικη, Bringer of Justice). In the Latin, Persephone gained another title: 'Juno Inferna', literally translated as 'Infernal Queen', likening her to Juno (Hera) as Her husband was likened to Zeus.

It's important to note that Perrsephone does not become Korê, or the other way around. Persephone's name is 'Persephone' (Περσεφονη) or a variation thereof, depending on author (Persephoneia, Persephoneiê, Persephassa, Phersephassa, Phersephatta, Pherrephatta). Her titles include 'Korê', and because she was well known as such, it was often used as a replacement of her name.