According to Akallabêth, Avallónë gained its name because it was 'of all cities the nearest to Valinor'. Actually the entire name comes from the older form Avallóna, the 'island near to Valinor' (originally an alternative to Tol Eressëa for the whole island, only later applied to its haven city). It should be noted that later texts imply that the name came instead from Avallondë, where londë means 'haven', but this appears to be a later rationalisation rather than the actual origin of the name.
It should be noted that in Tolkien's earliest tales, there was a layer of Arthurian myth to certain elements of the nomenclature, and Avallónë is apparently a remnant of this phase. Its similarity to the Avalon, the distant and mysterious isle of Arthurian myth, can hardly be coincidence (as indeed Tolkien indirectly hints in his correspondence).