Hi,
I wouldn't usually contradict someone when you have said you are pretty set on something but driving really isn't a good idea in my opinion. I live in Italy, just south of Rome and regularly drive to Cannes, south of France, and in the summer to the French Alps and Switzerland (a lot less then you are planning). viamichelin.com says London to Rome via Torino is 25 hours driving - that is without any hold ups, problems etc. So that will be an absolute minimum of 50 hours in the car over 8-10 days - more if you are coming from further north in the UK.
You will probably need at least 2 stops before Turin, then say 2 nights with your friend, then say two nights in Rome, before at least 2 nights on the way back. That is your 8 nights gone with only 2 days spent when you are not in the car on a long trip - you will probably be spending at least 6 or 7 hours in the car on each of those days, quite likely more.
Motorways in Italy, especially in the mountains, the stretch along the coast through Genova, and a lot of Tuscany are mostly two lane, with many tunnels - very scenic but a nightmare driving when it is busy (which is most of the time, they are full of lorries) and possibly in bad weather. You will also probably be driving in the dark some of the way. A car in most Italian cities is a liability - I live 20km from Rome and never go there in the car (and I am allowed to drive in any part of the city). If you do stray into a limited traffic zone, even for a minute (and they have them in other cities as well), the fine will be about 80 euros.
We drove to Cannes last week, due to bad weather, parts of the motorway being closed and other delays it took us 10 hours each way - a bit more than usual but not that unusual. We stopped 3 times going there and back but only for 15 minutes each time. If you are planning to go through the Mont Blanc tunnel you can expect a delay of 1 or 2 hours even on a not particularly busy day if you can't plan to get there at about 6am. You are probably looking at about 60 hours in the car over the 8 days without any significant delays - which are not that unusual. Worse case you could get stuck somewhere - the weather in the north of Italy has been horrendous the last few weeks, floods and landslides everywhere.
If you do go, it will cost you around 450 euro each way (from viamichelin), plus the cost of hotels, petrol here in Italy is very expensive. You will need to change to winter tyres and carry snow chains, that is just for Italy.
Sorry this has been so long and very negative, but seriously, take the plane - no way would I attempt that trip over 8-10 days in winter, with an 8 year old, and two cars trying to keep track of each other. Yes, you can do a lot with good planning but you can't make the distance any shorter and you can't make the roads better, and you can't move the mountains.