Nope. No storage at the station at all. People have posted here before that a shop called Oceans in Southampton High Street will hold it for you, but I have not investigated further. Or try the airport.
If you are exploring Southampton then do get a taxi to Oceans the only place other than hotels where you can leave your luggage. It is located in Southampton's High Street (Below Bar). www.oceanssouthampton.com Then start your exploration from the historical Bargate which you won't be able to miss outside the shop. Guided walks start from here or you can do your own - it is well-sign posted as "Walk the Walls" and takes visitors around the Medieval walls of the old town passing museums, good pubs and pass shopping centres. You can also book a guided walks via their website for specific topics such as History of The Titanic, Jane Austen Trail etc. most take about an hour. This site may also be of interest: www.discoversouthampton.co.ukEdited: 2:51 am, May 19, 2017
Update: I used www.citystasher.com. Twelve pounds for two bags. Expensive but better than lugging them around with me.