Non-volatile storage directly on the core?
Hmm, not sure about that one. We don't put much RAM on the core - most RAM in a system lives elsewhere, and its data gets pulled across to the cache as needed. The point of memristors is that they get you fast rewriteable non-volatile storage. But the point of a cache is that it's *NOT* a permanent store.
You can't use it instead of RAM, that's for sure. 10^12 write cycles sounds amazing - but run the numbers and you find that even if you only wrote to it at 100kHz rate, that would only give you 115 days of continuous use. So we still need the old architecture of volatile data living in RAM for processing and input/output data being written elsewhere at infrequent intervals for long-term storage.
IOW, it's a really good, fast, reliable non-volatile storage mechanism. Nothing more, nothing less. Currently Flash SSDs are a bit niche, but it seems likely that memristor NV will basically blow away Flash and hard drives for everything except really large storage application, following the same principle as tape drives being used for storage/backup when hard drives were the fastest available technology for small/medium data.
Sure, the idea of "full shutdown" will basically go away. With near-instant storage of system state for hibernation and recovery, that's all anyone will do. But this is already the case - most people will already hibernate their laptops instead of doing a full shutdown.
So it really ain't a game-changer. It might change the names on the storage offerings, but it won't change what we do with PCs, how we do it, or how the system architecture is set up.