J.J. Kneen included a couple of pages in the origin of (the name) Man in the introduction to his book (Placenames of the Isle of Man)published in 1925. He keeps the debate open but suggests that the earliest names applied in writing (early Latin from Caesar - Mona, Pliny - Monapia etc) with the root Mon were Pictish rather than Gaelic in origin and based on the Pict word for mountain Monadh rather than the Gaelic word Sliabh and suggesting linguistic and tribal links with Scotland and Wales rather than Ireland and England in late BC/ early AD times. My father and grandfather told me repeatedly that the island was Mona's Isle - but this confused me as it was also the ancient name for Anglesey!

It is suggested that through linguistic developments arising from trade links and tribal/national conquests over the centuries, the root Mon developed into Man and when various genetive or dative suffixes are added Mann. Examples of linguistic devlopments are:
Early Latin (54BC-100AD) Mona, Monapia
Later Latin - Mannia with genitive Manniae
Irish (1000-1400 Irish Annals) - Mana (similar sound to Mona) or Manu had the genitive form Manann and dative Manainn.
Welsh (1100-1200 - Welsh Annals) - Eumonia (1125), Manaw (1154)
Scandanavian (1240 Scandinavian Sagas) - Mon had genetive Manar

In summary, it seems that the Pictish inhabitants of the island in early historical times refered to the island as Man meaning mountain but sounding like Mon and written that way by Caesar. This developed into the genitive version in Mannan Eye/Isle (or similar) and became shortened to Mann or Man.

No doubt a cultural linguist will tell us different version.

So - you take your pick, depending upon what national/ political/ historical/ cultural message you wish to convey historically or modernly - as Manks (the true spelling!!!) people have done over the centuries!