The title is said to originate[1] from the town of Clare, Suffolk, which was owned by the first Duke of Clarence, Lionel of Antwerp. His wife, Elizabeth, 4th Countess of Ulster, was a direct descendant of the previous owners, the de Clares, and the Manor of Clare was among the lands which she brought to her husband.[2]
References:
1) Polydore Vergil, in his Anglica Historia of 1534 (Book XIX.36) dates the Dukedom to 1361 and claims to have rediscovered the lost origins of the name. See also David Hatton, Clare, Suffolk, an account of historical features of the town, its Priory and its Parish Church, 2006, Book 1, p21 ISBN*0-9524242-3-1 It is also available online on the Clare website. However, other writers, e.g., T.A. Trant, Narrative of a Journey Through Greece (London: 1830) p4, trace the title Dux Clarentiae to Clarentia (Glarentza) in the Peloponnese, seat of Matilda of Hainaut (aka Maud, Mahaut), Princess of Achaea from 1313-1318, from whom her cousin Philippa of Hainault might have passed to her son Lionel at least an honorific claim to Achaea.
2) One or more of the preceding sentences*incorporates text from a publication now in the public domain:*Chisholm, Hugh, ed. (1911). "Clarence, Dukes of". Encyclopædia Britannica. 6 (11th ed.). Cambridge University Press. pp.*427–428.