If I'm perfectly honest, if not altogether serious, I'd like to see all of them open. But recognising that probably isn't realistic - my criteria for which lines should have been kept open would be:
* providing direct links between large towns and cities which currently don't exist
* all sizeable "market towns" (places the size of Romsey, Alton, Wimborne, etc) should have kept their rail service
* large villages or small towns with a large commuter population to a large city (London, Birmingham etc) should have a rail service.
So in my area, I'd suggest - or agree if the suggestion has already been made - with the following:
* Romsey - Andover - Swindon via Stockbridge and Ludgershall. Rationale: no direct links between Southampton and Swindon or Cheltenham/Gloucester; no north/south links from Andover. "Regional Railways"-style service (158 or similar) Southampton to Cheltenham every 2 hours, using the existing line north of Swindon. Even though in an ideal world I would open all stations, a compromise solution would be to open only the larger places (Stockbridge, Ludgershall etc) to allow an all-stations service which is relatively fast for longer-distance journeys.
* The Somerset and Dorset (know this has come up before) Bristol to Bournemouth, with some sort of alteration at the north end to allow trains to get onto the main GW lines. For similar reason to above, limited intermediate stations: Radstock, Midsomer Norton, Shepton Mallet, Templecombe, Sturminster Newton, Blandford. Again, Regional Railways-type service, 158 or similar, again every 2 hours, with peak additionals Bristol-Shepton Mallet and Blandford-Bournemouth. Rationale: Bristol and Bath to Bournemouth direct services but also several stationless towns and a wide area with no stations.
* As an *alternative* to the Somerset and Dorset, how about Bournemouth to Salisbury, though the route the original line took was a bit odd. A better route would be direct up the Avon Valley serving Ringwood. Rationale: direct link Bournemouth to Salisbury, Bath and Bristol; station at Fordingbridge. Service: extend some of the London-Salisbury stoppers and/or Bristol to Westbury stoppers (via Warminster, Salisbury and this line)
* Castleman's Corkscrew, to serve Ringwood, Wimborne and the surrounding areas. Some sort of curve at the west end to allow trains to run back into Poole and Bournemouth which would be the most useful thing to do. Probably best to electrify to allow London, or at least Southampton, through services rather than a perhaps poorly-used branch. A suggestion which has come up before somewhere has been to reroute the xx39 out of Waterloo over this line beyond Brockenhurst, which would give a direct Southampton service or direct but slow London service.
* Guildford to Cranleigh. Cranleigh is a sizeable place well within London commuter territory. Really needs through services so electrify immediately and introduce hourly service (450 detached off the London-Portsmouth or London-Haslemere stoppers), half-hourly in peak hours.
* Would concur with Winchester to Alton though couldn't really do it now as it would disrupt the Watercress Line. Alton, Farnham etc are a PITA to get to from the Solent area, and extremely expensive given the as-the-crow-flies distance. Maybe a realistic proposal would be to turn the 64 bus into a "rail bus" (given it's run by the same people as the railway) and allow use via rail tickets, such that Southampton to Alton, for example, tickets would be valid on the bus and be priced similarly to say, Southampton to Petersfield, a similar as-the-crow-flies distance.
* Don't really know the routes of the lines, but places like Glastonbury and Wells could do with a train service, though it would probably have to be direct to Bath and Bristol to be viable.
* Did Bedwyn to Marlborough ever exist?
* Curves at Dorchester and at Yeovil to allow through Bournemouth-Exeter services, Bournemouth to the southwest must be a complete pain at the moment. Not sure if they ever existed though.
Purely fantasy, and didn't even fully exist back in the day, but I mention as it was my local area:
* Haslemere to Chichester via Midhurst, giving a more direct Chichester to London service as well as giving a rail link to an area which is something of a blank on the map. Topography would render it extremely difficult to construct, but as I said, purely a fantasy idea
