Is Simon's problem a good NP-intermediate candidate?
Simon's problem is not a pure NP problem, for two reasons:


It is an oracle problem.  We are given an oracle for some function $f$.  That's not something that you can do within the definition of a NP problem.
It is a promise problem.  We are given the promise that $f$ will satisfy a particular property (it is two-to-one, and has a particular structure).  That too is not something you can do within the definition of a NP problem.


So Simon's problem is not a problem in the formal complexity class NP; it's just something different.  For the same reasons, it's not NP-intermediate, either.