Tolkien says the name is '...derived from rare Middle English Westernesse ... where the meaning is vague, but may be taken to mean "Western lands", as distinct from the East inhabited by the Paynim and Saracens.' (From The Letters of J.R.R. Tolkien, No 276, dated 1965).
Whether intended by Tolkien or not, it is hard to avoid a connection between Westernesse and Lyonesse, the British 'Atlantis'. This was a region of western Britain that sank beneath the waves in Arthurian legend, and was supposed to be inhabited by mighty men of old (some of Arthur's knights came from there: Sir Tristram was probably the most noteworthy of these).
This Arthurian connection is not to be wondered at: Arthurian elements and themes often appear in Tolkien's work. The most obviously connected placenames, for example, are Avallónë (compare Avalon, and consider that the entire island of Tol Eressëa was at one time called Avallon) and Broseliand, an original name of Beleriand, which is also the name of a faery region in the tales of Arthur.
Whatever its source, the meaning is probably ultimately Old English westerne næs, where næs (modern ness) means literally 'promontory' or 'headland'. Tolkien was certainly aware of this (he used it himself in Taras-ness). Westernesse, of course, was not a headland but an island, but Lyonesse was a promontory, which hints further at an Arthurian connection.