As much as I like the schools of magic, I think they were better in 2e/3e where you specialised gaining bonus spells and losing access to others. For 5e, I think it would be much better to go with strongly themed archetypes. I'm not sure exactly what these subclasses should be, but something like:
- The Dark Mage, a wizard who deals with the dark arts or necromancy and demon summoning (perhaps not the best for a PHB handbook, but a great villainous option). Since subclasses are fairly small, this might also need to be split between necromancer and demon summoner.
- The classic Illusionist.
- The Elementalist. A wizard skilled with destructive elemental magic, perhaps focused on a single element, but skilled with all.
- The Arcanist. A wizard that really gets a lot out of their arcane implements. They might be the best ritualists or maybe they gain additional powers from using their arcane focus depending on if it is a wand, a staff, an orb, or a spellbook.