The Staff of Solomon was given to Solomon Kane by N'Longa, an African witchdoctor, who claimed he acquired it in ancient Judea. N'Longa can use the staff to communicate with Kane over great distances, and it has shown to be able to slay undead and mythical creatures.
In Robert E. Howard's story The Footfalls Within, a man known as Yussef the Hadji recognizes the staff for what it is. He claims the staff is older than the world itself and holds mighty magic. Its current appearance was also not its original form, as the cat-like head it is adorned with was carved as a representation of Bast by her priests in ancient Egypt; they also added the hieroglyphics engraved into the body of the staff. Moses used the staff to do wonders before the Pharaoh and carried it with him when his people fled Egypt. For centuries after it was known as the Scepter of Israel (mentioned in Numbers 24:17), which the prophet Solomon used to combat magicians and capture evil djinn.
Even before this, Atlantean priests in cities beneath the seas used the staff to fight evil, millions of years before mankind was even born.
The staff itself is made of a wood that exists nowhere on Earth, and is said to have in fact been grown in another world entirely. Presumably this is what gives the staff much of its power, with the Egyptian alterations to it being quite possibly just superficial; although the cat head of Bast does seem to carry a subduing nature to it when gazed upon.