Southampton's future is solely dependent on attracting airlines, the right routes, and providing them with incentives to operate from there. An extended runway does not guarantee that this will be the case.
The routes need to be ones that passengers want in sufficient number to make the operation economic, and the airport needs to both promote itself to the passengers and ensure that they have a great experience passing through the airport.
As has already been pointed out, LCC's will not base aircraft at an airport that restricts their ability to utilise each aircraft for three rotations each day. Southampton seems unlikely to get more than an away based aircraft operating on a W basis, unless the operational hours can be extended. Will the local residents accept this? Seems highly unlikely I suspect.