Tonni is a variant form of the Old Danish and Old Swedish masculine name Tunni, itself perhaps a shortened form of an East Scandinavian name *Thurniutr (von Feilitzen 1937: 386, 389; Fellows-Jensen 1968: 290, 293).
The relevant entries in the PASE corpus are for Tonni 1 and Tunni 1, which are catch-all categories for occurrences of these spellings in DB; both forms are treated here as representing Tonni. There are also two entries for the Old English name Tunna, which does not seem to occur in DB; both entries refer to men active before the eleventh century.
Fellows Jensen 1968: Gillian Fellows Jensen, Scandinavian Personal Names in Lincolnshire and Yorkshire (Copenhagen: I Kommission hos Akademisk Forlag, 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: Tona, Tone, Tonna, Tonne, Tunna, Tunne
Forms in modern scholarship:
von Feilitzen head forms: Tonni, Tunni
Phillimore edition: Tonni
Alecto edition: Tonni, Tunni