Is there any textual evidence in Hebrew Bible or Christian New Testament that would identify the Pharaoh involved with Moses and the Exodus? Is there any textual evidence that would also correspond the earlier Pharaoh (some 300 years earlier) for whom Joseph would have been the vizier?
The most common times chosen for the Exodus are sometime between 2670 BCE and 1759 BCE, during the Reign of Ahmose I who ruled 1539–1514 BCE, during the reign of Amenhotep II who ruled 1427–1401 BC, or 1427–1397 BC. Most scholars believe that it was one of the Pharaohs of the 18th Dynasty
The thinking behind these two dates are thus: For the first time period it is believe that slaves would have been needed in order to construct the Pyramids. The Hebrews are an easy choice for that. This does not enjoy much support however as there is no indication that there were slaves in Egypt and records tend to indicate that the pyramids were built by Egyptians.
For the second time period of the 18th Dynasty, there are several reasons for this choice. First, there are the Hyksos which literally translates to "ruler(s) of the foreign countries". Records indicate that prior the arrival of the Hyksos, Egyptians were darker skinned with African skin tones. With the arrival of the Hyksos, lighter skin tones such as those of Asian or Semitic decent appeared in the art of the Egyptians. This would be consistent with the Biblical claim of Joseph as a foreign ruler in Egypt. Furthermore, refrences to the Habiru, (probably the Hebrews) a nomadic people living in the fertile crescent did not appear until well after the time of the pyramids making the times of the pyramids a very unlikely choice for the Exodus.
The Habiru also appear in Egyptian writings noting them as slaves of Egypt during the reign of Amenhotep II.
The reasons for choosing Ahmose I as the Pharaoh of Exodus are that he was most known for chasing the Hyksos into the Sinai desert and waging war against the Hyksos which might be responsible for the note in Exodus 14:9 which indicates that the Israelites were overtaken by the Egyptians and an altercation occurs in the passages after this, though fighting is not mentioned in the Bible.
The justification for choosing Amenhotep II is that there is no record of the Hyksos as slaves under Ahmose I (though it is possible that the Hyksos were enslaved for a time.) Shae suggests that Amenhotep II may have died in an altercation in the Sea of Reeds which could account for Exodus 14:26-28
There are also several other options for the pharaoh of Exodus outlined here.
Indirectly there is text pointing to Menkaure.
Exodus 12:5 describes what the Passover meal should be. Genesis 32 states that Jews did not consume meat from the hind legs.
Richard Redding (an archaeologist at Egypt) analysed the bones at the pyramids and there is a website claiming that these bones match the description of Exodus 12:5 and Genesis 32.
Seals from this site bear Menkaure's inscription.