I think the driving will be fine - but you need to work out the timings carefully so that you don't hit the M25 at rush hour...
Look at the journey on Google Maps and work out how long it should take. Then work out when would be the best time to get onto the M25 (mid morning? lunchtime? early afternoon? late night?) and work back to when you need to leave.
And then do the same with the M3 and see if there's an obvious starting time for the journey which will allow you to do the entire journey avoiding the worst of the traffic.
When you're doing the journey, make sure that you have your sat nav with you and that you know how to use it while you're driving.
Also print out the details from Google maps for the trip and look at them too. Use the on the road view to look at all the motor way junctions you'll need so you can see what they look like in advance, whether there's a lane that splits off, a big roundabout at the top or a turning or whatever.
Make sure you have some drink with you that you can easily drink safely while driving, likewise snacks that you can eat as you go (both easy ones like sweets or chocolates to pop in, a bag of grapes that have been picked off the stalk and washed and popped into a bag, cucumber chunks in a bag etc) and things like bananas or museli bars that you can get energy from and eat if you're stuck in a traffic jam or at a service station but not that you'd eat as driving.
Then you should be fine...
Also, have you really played around on a site like the Train Line to check out fares? I've just had a look and while if you wanted to go this week there aren't many cheap fares around, if you wanted to go in a couple of weeks and could commit to specific trains rather than have a ticket that would work on any train on the day, then you can go much more cheaply.
I looked for a couple of weeks time - there was a single from Norwich to Weymouth (have just chosen a big town from each county!) for £34 going - and coming back on a friday, there were only standard singles to come back for £99 but they had a few £45 first class singles left - which would mean that it can often also include food, drinks, newspaper, free wifi, nicer waiting room, depending on your train company so it might also be worth upgrading to first class for the outward journey too at that price as you'd save by the time you'd added those in (just check on which train line offers what - I know it varies according to line but that's what I'd get on my local line). And friday's often an expensive day to travel...
So if you have flexible dates in the future and you do fancy going by train, it would be worth working out how long a drive you think it would be and seeing how much you think it would cost in your car - because you might find that there's not that much of a difference... looking at the journey on //www.viamichelin.co.uk and using their very basic journey calculator for the same trip in a hatchback, putting the petrol cost at £1.15 a litre, it comes in between £29 and £35 (I know it will vary depending on what car you have!). And that's without including anything for wear on your car. Plus on the train you don't have the stress of driving so can treat it as part of the holiday if you have a nice comfy seat and book etc... So the costs needn't be that different if you did go by train too.
Oh and I've just remembered split ticketing - check out the money saving expert tool (www.moneysavingexpert.com/split-cheap-train-tickets/) (I expect there are others too) or even just look yourself at the two halves of the journey...
I just looked at Norwich - London (cheapest single a bit in advance - £9) and London - Weymouth (£14). You'd need to add a single tube fare across london (£2.40 if you have contactless) or get it built into one half of the ticket. And you'd need to make sure that you built in plenty of time to cover delays as your two tickets would be independent, unlike if you bought a straight through one (when they'd have to find you a seat if your other train was running late and you missed the connection). So you'd be able to do the lot for about £50 return... which sounds a lot more reasonable than the £100 off peak standard return!