Have just googles "Mayflower terminal left luggage" and came up with the following.
Are there any left luggage facilities in Southampton?
In the present security situation, there are no conventional left luggage facilities. There is a self-storage company, Lok'nStore, which allows passengers to leave pieces of luggage at their warehouse in the Millbrook region of the city. Please see their web site, or telephone 023 8078 3388 for further details. The airport has offered this service in the past, please telephone the Information Desk on 0870 040 0009 for more information.