I'm surprised no one has suggested running HS2 under central London. If we look at the future gaps raised in the London and SE RUS these are suggested to be dealt with by tunnels from Wimbledon and Purley for fast trains. The former feeds in to Crossrail2, no details on where the latter might go but it would have issues adding more passengers for dispersal at a terminal like Victoria.
My suggestion is that HS2 has a 4 platform station at Euston heading into a tunnel with a more 4 platform stations at Waterloo and East Croydon before emerging in Purley. A re-engineered route to Gatwick Airport passed for continental gauge is used to reach a suitable terminus station at Gatwick (8 platforms?) some CC trains would continue on down the newly OHLE converted Brighton Main line to serve Haywards Heath and Brighton.
This solution solves many issues including space at Euston, dispersal issues and south London capacity problems.
Tunnelling under London has been mentioned previously either just as far as Euston, or right through London. The former is favoured by Boris Johnson and London councils on the route and not favoured by HS2 ldt because of the extra cost. Some members have suggested tunnelling through London, but the usual criticism is that it's alot of expensive infrastructure that will only be used by a minority. As I understand Crossrail 2 would divert of the swml at charing cross and take them to Euston. You are however the first person I can remember suggesting running hs2 to Gatwick after Euston. My initial thoughts are that Gatwick is cuuently only a one runway airport and whilst it could be enlarged to 2 runways it is unlikely to be the largest airport in the southeast, where some form of high speed connection should be directed.