Who was Pharaoh when Moses lived in Egypt? Do we have a good idea as to which Pharaoh(s) were around during the time of Moses' life? What internal and external evidence exists to back up those claims? Do any Christian traditions attach any significance to correlating the events of Moses' life to the reign of a specific Pharaoh?

The movie The Prince of Egypt mentions Ramses as the name of Moses' adoptive mother's son (and the future Pharaoh, both when Moses kills the Egyptian at 40 and when he returns to Egypt when he is 80), but I don't know what the basis is for this claim.
Of course this is more of a historical-archeological question as the Bible does not directly answer but I have stumbled across a fairly convincing argument that would place Thutmose II as the Pharaoh at the time of the Exodus.

I first came across the argument in Alfred Edersheim’s Bible History. He typically pays attention to ancient monuments and secular history, as he was a historian at Oxford. He was also a Jewish man and Jewish historian converted to Christianity which makes him the most reliable source of Jewish culture and history at the time of Christ.

In his book he argues that all the records of Thutmose II best fit into the Exodus narrative:


  Now this Thothmes II. began his reign very brilliantly. But after a while there
  is a perfect blank in the monumental records about him. But we read of a general revolt after his death among the nations whom his father had conquered. Of course, one could not expect to find on Egyptian monuments an account of the disasters which the nation sustained at the Exodus, nor how Pharaoh and his host had perished in the Red Sea. But we do find in his reign the conditions which we should have expected under such circumstances, viz., a brief, prosperous reign, then a sudden collapse; the king dead; no son to succeed him; the throne occupied by the widow of the Pharaoh, and for twenty years no attempt to recover the supremacy of Egypt over the revolted nations in Canaan and east of the Jordan. Lastly, the character of his queen, as it appears on the monuments, is that of a proud and bitterly superstitious woman, just such as we would have expected to encourage Pharaoh in "hardening his heart" against Jehovah. But the chain of coincidences does not break even here. From the Egyptian documents we learn that in the preceding reign - that is, just before the children of Israel entered the desert of Sinai - the Egyptians ceased to occupy the mines which they had until then worked in that peninsula. Further, we learn that, during the latter part of Israel's stay in the wilderness, the Egyptian king, Thothmes III., carried on and completed his wars in Canaan, and that just immediately before the entry of Israel into Palestine the great confederacy of Canaanitish kings against him was quite broken up. (Alfred Edersheim’s Bible History, Vol 2)


What adds striking support of this argument is just this year an online article was published saying a ‘Harvard University educated archaeologist and president of the Paleontological Research Corporation, Dr. Joel Klenck, states an array of archaeological discoveries evidence a crisis during the reign of the Egyptian Pharaoh Thutmose II’. You can find the find the article here.

In this article it is mentioned that:


  The mummy of Thutmose II is the only corpse of a pharaoh during the Eighteenth Dynasty covered with cysts from an unknown malady. (Paleontological Research Corporation, Dr. Joel Klenck)


Possibly his body was fished out of the sea and mummified and that the ‘cysts’ were part of the plagues/boils which he had suffered?  This article is really worth reading as it makes a strong case that fits the Moses narrative.  The article does not mention the Bible or Moses, but I find it hard to believe the author is not making the implication but for some reason not stating it.