One explanation
The origin of the name Bosomworth is lost. My own preferred explanation is that the name comes from the place name Bosham, a village in Sussex noted for its connection to King Harold and the events of 1066.
The local Sussex people pronounce the name Bosham as 'Bossum'. "Worth" is derived from the Saxon meaning 'a court, farm, possession, place, field or way; the place valued, sold, or granted.'
Putting these together might mean Bosomworth refers to a person owning or originating from a farm or lands around the place Bossum. Spelling was not an exact science in those days, which explains the variations we see in records.
The family appears to have flourished in and around Yorkshire from the available parish records. The Norman invasion may have displaced some of the Saxon Bossumworths northwards.
An alternative origin sometimes suggested for the name.