If that's £100 each then that's not great, but it's a bit late for Advance Purchase. That can't be £100 for the group, surely?
The cheapest non-Day off peak return ticket from London to Southampton is routed "Southern Only" and is £56.00 for a group of 4, using Group Save. (In Avantix Traveller NFM05 search for: Origin: 1702. Destination: SOU. Discount: GS4. Adults: 4). This works out at £14 each.
An Off Peak return from Hartlepool (HPL) to London is £69 per person on Grand Central only.
These tickets are all available on the day, and in theory all can be bought on GC (but GC may have difficulty with the Group Save), and is a total of £83 each.
This is far better than using the awful, overcrowded Cross Country route operated by cramped 4-car DMUs that rattle and stagger zig-zagging over the country acting as inter-urban trains between neighbouring towns and cities. And cheaper than the through "Not London" ticket priced by Cross Country, at £113.40 (a rip-off).
Hope that helps, and enjoy your trip on a decent train!
(If speed is an issue, then it's £70.20 for 4 people from Waterloo to Southampton on South West Trains using Group Save 4, this is £14.20 extra for the group, or £3.55 extra per person)
You will need to get across London. For 4 people you could consider a taxi. If you want to take the tube, then don't pay rip-off LU cash fares of £4 each way per person (total £32!), instead either use Oyster PAYG or buy a ticket from Finsbury Park to Southampton on Group Save, this is £76.80 for the group (£19.20 per person) and includes the Underground and is valid on SWT or Southern to Southampton.