Check the pitch
Some extension roofs have been built at a shallower pitch than the tile or slate really wants. Water can then be driven back under laps in bad weather. If a leak has been there since the extension was built, this is worth considering.
Wall abutments
The flashing where the roof meets the house can crack, pull out or simply have been formed badly. Render can hide the top edge, which makes the fault less obvious from below.
Roof windows and penetrations
Anything that interrupts the covering needs a proper drainage detail around it. Roof windows have their own flashings; soil pipes and vents have collars. A stain near one of these should be traced rather than guessed at.
Do not forget the gutter above
On some extensions the main roof gutter discharges close to the new roof. An overflow there can send a lot of water onto a detail that was never designed for it.