How did Egyptian Pharaohs get their title?

In the Book of 14 Jasher there was a man from Shinar Ancient Sumer who had wisdom and understanding. His name was Rikayon he was indigent and poor and hard set to support himself. He went to Egypt to Oswiris or Osiris the son of Anom King of Egypt to inquire about employment but the King was not in because it was his custom to leave the palace and travel abroad one day a year and hear law suits.

Rikayon stayed in Egypt living in a burnt out bakery waiting for the king to return. He tried to become a vendor and sell vegetables but he was ridiculed for his technique. So the next day he determined to go to the Egyptian Sepulchre and tax the people who wanted to bury their dead 200 pieces of silver. He became wealthy from this. The king was angry when he heard of it, but Rikayon plied him with gifts and when Osiris heard the the story, Rikayon found grace in Osiris's sight and Osiris told him you shall no longer be called Rikayon but Pharoah shall be your name. The king and his subjects loved Rikayon and determined that all kings over the land of Egypt from Osiris forward would be called Pharoah.

This is from the Book of Jasher 14 1-33

I agree with Sempaiscuba's answer. I do not think the events referred to in the question are historical and can only add snippets of related facts.

Going by such Egyptology and basic hieroglyphics and Ancient Egyptian language as I have read and studied (mainly Manley & Collier How to read Egyptian Hieroglphics and Manley Egyptian Hieroglphs for complete Beginners) the normal Egyptian words for the King of Egypt on Egyptian Monuments certainly down to the end of the New Kingdom were: 'Nsw' (their script normally only wrote consonants so it is customary to pronounce it 'Nesoo'. For simplicity I shall use only pronounceable forms of words below) which meant the King of Egypt in his religious role and 'Bity' (represented by a bee hieroglyph) meaning the king in his secular role.

Among themselves in Egyptian they referred to foreign kings as 'Wer', the same title given to important non-royal Egyptians like provincial governors. This implied that all foreign kings were rightfully subject to the king of Egypt, who was the only proper king. For the purpose of external relations the Egyptians generally wrote in other languages, in which they were more diplomatic.

Egyptian Kings also used the titles 'Netjer Nefer' (=the good god or perfect god) and 'Neb Tawy' (Neb meaning Lord and Tawy being the 'dual' form of the word for land, so 'Lord of the Two Lands') which referred either to Upper and Lower Egypt i.e. the Nile Valley and Nile Delta or perhaps to the East and West banks of the Nile.

'Per', a masculine noun, was the word for house. 'Per aa' meant Great House and began to be used as a term for the King's Court, which is the origin of the Hebrew biblical word 'Pharoah'.

I do not recall anyone called 'Rikayon' or 'Nimrod' being mentioned in the Egyptology I have studied.

Nimrod is the name of a great-grandson of Noah mentioned a few times in the Bible, especially Genesis Chapter 10 verses 8-10. We are told little of him for definite except that he was 'mighty' and a 'mighty hunter'. Hence the composer Sir Edward Elgar gave the name 'Nimrod' to the best known of his 'Enigma Variations' in honour of a friend of his called 'Jaeger', which is the German word for 'hunter'.

The short answer is: "No, there is no evidence of this in historical records".

The word, 'pharaoh' [pr aA] is first attested in the First Dynasty, about 3150 BCE. It means 'great house', or 'palace'.

It wasn't used as a title by Egyptian kings until the reign of Thutmose III in the New Kingdom (his reign lasted from about 1479 to 1425 BCE). From this point, the title 'Pharaoh' becomes the norm for Egyptian rulers.

Interestingly, Thutmose's step-mother, Hatshepsut, who preceded him as ruler of Egypt, is often referred to as 'the female Pharaoh', by modern authors, but - as far as I'm aware - no contemporary text or inscription includes 'Pharaoh' among her titles.

Nimrod, on the other hand, is a figure mentioned in the Bible, but one who probably didn't actually exist as an historical figure. Even in the Bible, he is not an Egyptian king, but a king in Mesopotamia. As the Wikipedia article observes:

Attempts to match Nimrod with historically attested figures have failed. Nimrod may not represent any one personage known to history, and in reality is more likely a conflation of several real and fictional figures of Mesopotamian antiquity.

As for the story of Rikayon, a character supposedly mentioned in the Book of Jasher, that may be a question better asked on Biblical Hermeneutics:SE.

However, there are a number of translations of later versions of the text available online, such as this example of the Book of Jasher from a book written in from 1613, if you with to wish to follow up on the stories.