southampton is close enough to London that I wouldn't suggest a car. Trains are hardly ever late, and are very cheap if bought in advance.
that said, i've had a few times where public transport (ie the tube, bus, etc) is very late, because delays due to signal failures or faulty trains, has made me a few minutes late for a train, and forced me to buy another ticket at full price, even when i had planned to get to the train station a good hour before it was meant to depart. (sorry run on sentence)
there's definitely something to be said about driving, as it's so much more flexible. I've been stuck in traffic in London though, and it's not fun.
£60 for 6 days is a pretty good deal. just remember that if you actually want to venture into central london, there's the congestion charge during the day, which is £8 a day m-f. if you picked up your rental outside the congestion zone, your car rental company will not cover the charge.
as well, parking in london can be an absolute nightmare.
if you're ok with all that, have fun driving!
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