Alaf is an Old Danish form of the Old Norse masculine name Áleifr and, like the side-form Óláfr and the archaic form Anláfr, comprised the elements *anu ‘ancestor’ and leifr ‘inheritance, son’ (von Feilitzen 1937: 143; Fellows Jensen 1968: ciii-civ, 6; see also discussion under head-forms Onlaf, Olaf). A normalized form of ODan Ālāf is used here for consistency with the other head-forms.
The only entry in the PASE corpus is for Aleifr 1, a catch-all for occurrences in DB.
Fellows Jensen 1968: G. Fellows Jensen, Scandinavian Personal Names in Lincolnshire and Yorkshire (Copenhagen, 1968)
von Feilitzen 1937: Olof von Feilitzen, The Pre-Conquest Personal Names of Domesday Book, Nomina Germanica 3 (Uppsala: Almqvist and Wiksells, 1937)
Spellings in Domesday Book: Allef
Forms in modern scholarship:
von Feilitzen head forms: Áleifr
Phillimore edition: Aleif
Alecto edition: Aleifr