"It’s what they used to say about me at school. Delphini Diggory - there isn’t a hole she couldn’t dig herself into."
-- Delphi (CC1.6)
The name Delphini is related to the name Delphina, which means "womb". It comes from the Latin boy's name Delphinus, which meant someone from Delphi, a city in Ancient Greece. This in turn may be related to Greek δελφυς (delphys) "womb" (Behind the Name). It could also be a shortened form of the flower name delphinium, which are beautiful plants but are "toxic to humans and plants" (Wikipedia).
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Pottermore: Harry Potter and the Cursed Child
The theatre production opened in the West End of London at the Palace Theatre on 31 July 2016, with Esther Smith in the role of Delphi (Delphini Diggory): Harry Potter and the Cursed Child, Parts One and Two