I can think of one connection, which is the homeopathic remedy, Nat Mur. That is made from sodium chloride, obviously in minute proportions because that is how homeopathy works. Nat Mur is one of the essential remedies in homeopathy for a combination of physical and emotional problems.

An odd idiom but idioms are odd.

The Latin word sal (salis is the genitive) means both "salt" and "wit", thus the Latin phrase cum grano salis could be translated to either "with a grain of salt" or "with a grain (small amount) of wit", actually to "with caution"/cautiously.[3] — wikipedia: Grain of salt

Apply a modicum of wit/caution toward whatever comes your way.

I'll have the corned beef.