Best way from Paris to Southampton port

You could fly to Southampton from Orly Airport on Easyjet

As mentioned, you could fly from Paris or Cannes to Southampton .

You could also train from Paris, with a change of trains and station in London.

There is no flight from Cannes (not an airport with scheduled services) to Southampton ever and the one from Paris Orly only starts on 31 March- and then only on a Monday and Friday, both days at 1850.

Any other day it is Eurostar to London St Pancras, then South Western Trains from London Waterloo.

Air France has flights from Nice to LHR, BA to LHR and LGW and Easyjet to London Luton and LGW, also Ryanair to London Stansted (seasonally)

Options look quite complicated! What about driving rental car? Would require drop off in different country which sound expensive.

It doesn't sound expensive. It IS expensive. Very expensive.

And will you drive a French car in the UK? Could be a challenge though I did drive with a Briton who drove us from Dijon to Nice in his British roadster and he was fine though to be fair, he lived half the year in Burgundy and half the year in London so he was quite used to the switch.

Your car drop off fee in the UK could be as much as $1000.00. The train sounds like your best option. Eurostar to London, then a train from London to Southampton, after change from St. Pancras to Waterloo station. Less complicated than the air options, IMO. For ease of transfer to Waterloo with luggage, I would take a taxi.

If you fly into Heathrow there is the National Express bus direct to Southampton.