I think that any plans to discontinue flying at Bournemouth would be a huge mistake. I am generally a supporter of SOU, as this is my nearest and most convenient airport.
However, the Eastleigh area to the north of Southampton is desperately short of development land, and indeed Eastleigh Borough Council are looking to bring forward a very contentious new development of around 5000 new homes on environmentally sensitive green belt land to the north of the town.
The Solent region as a whole has grown enormously in the last 20 years or so, and the region including Portsmouth, Southampton, Bournemouth, Fareham, Havant, Gosport, Eastleigh, Christchurch, Bournemouth and Poole now forms an almost contiguous urban area with a population of around 2 million, with a very high proportion of high earners.
The sensible approach would be to properly develop Bournemouth airport, with improved and increased parking, further enlargement of the terminal, a new link road to the A388 and a reliable 20 minute frequency bus connection to Christchurch rail station (re-named Christchurch and Bournemouth Airport).
The idea of calling the airport Bournemouth is too parochial, and this should be changed to South Central or Southern England or similar and all SOU flights together with new routes should be centred on Bournemouth to give the whole region a much-needed large regional airport allowing proper connectivity to this major UK region.
The existing site at Eastleigh airport could then be closed and sold for development.