I believe the hotel started as the Ballaqueeney Hydro. The Kelly family to which Mr. Walter belonged being associated with the nearby farm of that name.
The Ballaqueeney bit became 'Balqueen' then 'Bayqueen' under different ownership.
There seemed to be a vogue for Hydropathic establishments, as the Grand Islend Hotel in Ramsey was known to older residents as "The Hydro".
THere was I think a Hotel on Douglas Prom of the same name.