It's not easy being DC

The Southern Railway and its predecessors were early adopters when it came to electrification, and most of the former Southern region is currently electrified to 750vDC, third rail. However, third rail electrification is considered outmoded and most lines electrified since the 1960s have been with 25kvAC overhead wires. This combination of systems is now causing a lot of head-scratching when it comes to filling the gaps in electrification in the South East. North of the Thames, there are a number of places where 750vDC meets 25kvAC, and trains need to be dual voltage. The London Overground is an obvious example. The Gospel Oak-Barking line is currently being electrified to 25kvAC, which is sensible, as it carries a great deal of freight destined for the 25kvAC West Coast main line, but many of the other Overgound lines are third rail DC.
South of the Thames, there are only a handful of dieselised lines remaining. One is the Wealden line to Uckfield. It is surrounded by third rail lines, but the Wealden line campaign want it electrified with overhead wires, to connect to a proposed line to Croydon. This seems very ambitious, and I would say a more realistic option would be to electrify the existing line with 3rd rail and possibly rebuild the Wealden line to Brighton (which would run through the currently preserved Lavender line).
Another feasible candidate for 3rd rail electrification is the North Downs line from Reading to Gatwick Airport. Parts of the line are already electrified to 750vDC, so it would make sense to use this system for the rest of the line.