You will find it cheapest to get a ticket from Portsmouth Harbour, route valid via Southampton, BUT NOT LONDON.
Also try Southampton-Cardiff, Cardiff-Chester, Chester-Liverpool ... (split ap tickets, around £80 rtn).
or
Southampton-Birmingham NS-Liverpool LS ... (split ap tickets around £80 rtn).
You might also find that buying a ticket to a station in the Merseyrail area, for example, Sandhills, does not neseccarily get you it any cheaper, but means you can get the local network round Liverpool. Southampton to any station on the Merseyrail network is the same as Liverpool STNS.