What I think 2nd edition would benefit from (arranged vaguely in order of importance):
1) Sanity/Stability rules rewritten for clarity (probably based on last chapter in Keeper Resource Book)
2) Extended description of skills, with sample spends and historical overview (also from Keeper Resource Book).
3) Extended Keeper Section with more focus how GUMSHOW works, how to avoid common pitfalls, how to design scenarios based on common complaints and misunderstandings (possibilities of failure, non-linear scenarios, etc.)
4) A different introductory adventure, more friendly to the first-time GUMSHOE Keeper. The current one has some neat ideas, but requires a lot of improvisation/additional work on the Keeper’s side and some real leaps of logic on the players’ side.
5) Some new rules, adapted from later iterations of GUMSHOE, like extended contents rules from NDA.
6) Overall rules rewritten for clarity and fitted with more examples.
7) New Mythos creatures illustrations (I absolutely adore Jerome’s faux-photo pictures, but I think that most images of Mythos creatures sort of fall flat).
Perhaps another adventure, specifically in Purist idiom.
This is, of course, the maximum mission and I realize that including ALL of these changes is not possible, but I hope to see at least some of them.