What internal evidence supports the historicity of Abraham? Is there any internal or small external evidence of Abraham's existence?
I have heard a claim from skeptics that Abraham is just flat out invented figure who never existed and even COULDN'T have existed.


  Not only has archaeology not proven a single event of the patriarchal traditions to be historical, it has not shown any of the traditions to be likely. - Thomas L. Thompson, The Historicity of the Patriarchal Narratives: The Quest for the Historical Abraham (BZAW 133, De Gruyter, Berlin and New York, 1974), page 328


Is this valid? Note that I am not asking for extra-biblical mentions of Abraham. I am realistic, he was unimportant nomad in illiterate world. However, I am asking, is there any evidence for events and places and customs described in Genesis? Is there any internal evidence in the text itself like criteria of embarrassment or something that must be historical? I have always seen Abraham's life as too simple and non-embellished and non-fantastical for him to be made up. Beyond that, I never saw any other evidence for his historicity.
Internal evidence from the New Testament
This is a Christian site, so a Christian defense for the historicity of Abraham can be based on what Jesus or the apostles say about Abraham.  If Jesus / the apostles implied Abraham exists, then there's your internal evidence from the New Testament.
Some examples:

Jesus's genealogy in Matt 1:17.  Christians believe that Jesus is 100% human and in His human nature He is a natural descendant of Abraham

Jesus's claim that he existed before Abraham (John 8:58)

Jesus said that many Gentiles will have a feast with Abraham in the Kingdom of God (Matt 8:11)

James used Abraham as example of faith in the sense that we can emulate a REAL person from the past (James 2:21-24)


External corroboration from Ancient Near Eastern research
While we have not found a non-Biblical text / inscription naming Abraham, there has been a lot of progress in Ancient Near Eastern research in the past 100 years fueled by new discoveries in the 20th century which both Biblical and secular scholars used to correlate elements of the Genesis story with non-Israelite texts, customs, and material remains.
To answer the claim of Thomas L. Thompson you quoted I offer this excellent essay by Martin J. Selman: &quot;Comparative Customs and the Patriarchal Age&quot; published by IVP in a 1980 book &quot;Essays on the Patriarchal Narratives&quot;, listing the current (yet) uncontested correlations between ANE research and Biblical text after reviewing 3 waves of scholarship: 1) Consensus, 2) Destruction of Consensus, 3) Current Discussion.  The claim by Thomas L. Thompson was covered in section 2 (Destruction of Consensus), but the essay responded in section 3 with a review of later research that tempered his rather negative assessment.
Another value of this essay is in providing believers with a level-headed and a responsible scholarly (non-polemical) guidelines for comparison (section 3.4 to 3.6).  This quote is from section 4 (Conclusion), emphasis mine:

When the biblical and nonbiblical material is subject to proper control, the way is still open for the social customs of the patriarchal narratives to be legitimately illustrated and supported from a variety of historical contexts in the ancient Near East. The following list includes those examples which remain valid in the light of the conditions for comparison discussed above.
... (skipped: list of 12 currently uncontested correlations)
Since the large majority of these examples show that the patriarchal customs can be compared without difficulty with a wide range of material from the ancient Near East, it may be concluded that the patriarchal narratives accurately reflect a social and historical setting which belongs to the second and first millennia BC. More precise dates must of course be derived from other considerations, but neither van Seters' preference for first millennium material nor Thompson's assessment of the essentially nonhistorical character of the narratives can be supported by the evidence of the social customs. From the independent viewpoint of the historian, therefore, the social parallels make the historical existence of the patriarchs more likely.
But our conclusion also has hermeneutical and theological implications. Two examples must suffice. That Laban should conclude the details of Jacob's marriages twenty years after the marriages were agreed (Gn. 31:50), even though a marriage contract would normally have been drawn up at the beginning, is further evidence of his duplicity and of the difficulties with which Jacob had to contend. By contrast, Abraham's refusal to take a second wife or a concubine of his own in the face of Sarah's continuing infertility gives a new insight into his regard for Sarah's position as well as his faith in God's provision of the promised heir. A proper appreciation of the social and historical dimension of the patriarchal narratives thus leads to a more accurate understanding of Genesis' theological contribution, as in the specific application of Abraham's faith.