This is a topic I've looked into - in fact, I wrote to Network Rail when their new electrification plan was released last year under their utilisation plan. They sent me a glossy brochure back with all the facts written in it.
First of all, the most likely Southern candidate for conversion - WERE IT TO EVER HAPPEN (an important point!), is the Basingstoke area - both down to Southampton, across to Salisbury and up to Reading which would then connect with the additionally proposed GWML electrification which would be complete by then (if, of course it happened at all).
This poses a couple of issues though.
1 - stock for Basingstoke, Winchester, So'ton and beyond would need to be dual voltage, transformers, pantographs fitted etc... This would mean slightly less flexibility in rolling stock use and transfer, but not too difficult - SWT could easily, I'd imagine, cope with this given the resources and money to manage it.
2 - The debate of dual voltage lines - it's most likely that if OLE was put up, the 3rd rail would be lifted. Having both DC and AC electrification running alongside each other poses all kinds of earthing and EMC problems for the surrounding environment and the infrastructure. It only works in very short stretches (like between Farringdon and City Thameslink, or the lines out of Euston for a mile or so). SO all the way between Basingstoke and So'ton (a distance of about 30 miles) would pose challanges. You couldn't run wires all the way into waterloo that's for sure. Basingstoke is realistically the furthest towards London that OLE could or would work because of operational restraints, infrastructure restrictions and of course... can anyone here imagine what Clapham Junction - Queenstown Rd would look like if that lot had wires over the top?!
3 - clearances for the overheads on bridges and tunnels not designed for them - an old problem...
In reality the southern network is so deeply aligned to 3rd rail traction, it's unlikely to be changed in any of our lifetimes apart from, at the very most, the above Basingstoke area upgrade (and even that is a stretch) - unless NR has a sudden rush of blood to the wallet and rips the whole lot up and replaces the whole lot with OLE... but as someone else pointed out - WHY?
Let me get this straight - I'm no fan of 3rd rail. I have always seen it as "overgrown underground" traction - not proper main line use, but having experienced it for real, it's got some advantages:
1 - it's robust - strong winds aren't a problem. Even in the snow of last year, trains on the SWML ran very well (with a lot of sparking)
2 - It's cheap(er than 25kV overheads)
3 - The view from the train window isn't ruined by masts every 20m or so. This is also a plus point for all the NIMBY's who live alongside the track.
The negatives are really only two:
1 - It is power restricted, therefore, maximum speeds are limited to around what they are now (and have been for some time), but given the relatively short commuting routes that MOST of the 3rd rail network covers, this isn't a huge issue. The acceleration of a 444 or 450 equals anything anywhere on UK main lines and topping out at 100mph isn't exactly shabby is it?
2 - Safety - 3rd rails are generally considered more dangerous than overheads - not because of the increased current or that it's DC instead of AC (which can be nastier on contact with a person) - it's simply much easier to accidentally come into contact with - particularly at stations or level crossings where the general public have easy access therefore the likelyhood of an accident is higher.