Is there evidence to suggest that Abraham and Hammurabi influenced one another?

In a word, no.

Abraham and Hammurabi lived in different centuries, almost half a millennia apart.

I do believe Abraham existed. Jesus said "Before Abraham was, I am" John 8:58.

First, in this statement Jesus claims his eternal existence as God, the Son; and secondly it reveals his belief that Abraham was a real person not a myth.

Then Jesus, to show the Sadducees that life after death is taught in the Old Testament, spoke of Abraham, Isaac and Jacob as being still alive (as to their souls) in Matthew 22:23-32 and parallel passages Luke 20:27-38 and Mark 12:18-27. He is quoting Exodus 3:6 where God wants to tell Moses about himself saying "I am the God of thy father, the God of Abraham, the God of Isaac, the God of Jacob". These are just two examples showing that Jesus believed in a real Abraham.

To see how Biblical events and people interrelate with people and events outside of the Bible you have to have a Biblical chronology first. That is the great underlying need.

For me the great teachers of Biblical chronology are Edwin Thiele, Rodger C. Young, Carl Olof Jonsson ("The Gentile Times Reconsidered", 1986, Commentary Press Atlanta). Carl's work is limited to the era of the Babylonian Captivity. Eugene Merrill ("Kingdom of Priests", 1987, Baker Book House) builds on the work of Edwin Thiele.
Gleason Archer's "A Survey of Old Testament Introduction" is a wonderful work too.

For a useful website see www.rcyoung.org/papers.html

Edwin Thiele shows the date of 931 BC for the division of the kingdom on the death of Solomon ("The Mysterious Numbers of the Hebrew Kings",University of Chicago Press, 1951). From this, a chronology can be constructed giving the date of the birth of Abraham as 2167 BC.

See 1 Kings 6:1 and Exodus 12:40 to arrive at a date for the entrance of Jacob into Egypt of 1877 BC, when Jacob was 130 years old (Gen 47:9).

So Jacob was born 2007 BC. Isaac was 60 when Jacob was born (Gen 25:26). And Abraham was 100 when Isaac was born (Gen 21:5). So Abraham was born 2167 BC.

So Abraham was several centuries before Hammurabi.

With this framework, the Exodus from Egypt happened on Passover 1446 BC. An independent witness to this date exists in Ezekiel 40:1, understood in conjunction with Leviticus 25:9, as mentioned in Seder Olam and as revealed by Rodger Young. So we have two independent witnesses for the date of the Exodus:

Witness 1. The data in 1 Kings and 2 Kings for the reigns of the kings in the divided kingdoms. That data was like a secret code for centuries. The code was finally deciphered by Edwin Thiele in Mysterious Numbers of the Hebrew Kings, 1951.

The important point here is that it was like a secret code. That was providential. If it hadn't been like a secret code then it would have failed to have been possible to be an INDEPENDENT witness. In that the meaning of the years of the reigns of the kings was unfathomable for at least 20 centuries until Edwin Thiele's work the data can be considered independent from the next witness:

Witness 2. Ezekiel 40:1 and the comment in Seder Olam.
Seder Olam is a chronology compiled in the second century by a Jew who wanted to try to give as accurate picture as he could of the knowledge about chronology in the Old Testament era. One thing he says is that the jubilee in the years of Ezekiel was the seventeenth. (An English translation of the Seder Olam is online.) Ezekiel 40:1 is saying the tenth day of the month was Rosh Hashanah, New Year's Day. (In fact it uses the term "Rosh Hashanah" in the Hebrew text.) Usually New Year's Day happened on the first day of the month. Rosh Hashanah only happens in the tenth day of the month, on the Day of Atonement, in a Jubilee Year (Leviticus 25:9). ((A Jubilee Year was a special year every 49 years in the Old Testament era.)) This was a full 14 years after the destruction of the city of Jerusalem which happened in 587 BC. And Seder Olam says the Jubilee in the days of Ezekiel was the 17th. 587 - 14 = 573. 17 jubilees times 49 years is 833 years. 573 + 833 = 1406, the date of entry into the promised land. 1406 - 40 years wandering in the wilderness is 1446 bc, the date of the Exodus, which had already been arrived at by Edwin Thiele. See Rodger Young's article on his website.

Many modern scholars say that the Jubilee system was a late addition to the Jewish religion, added about 500 BC. But Ezekiel 40:1 and its interpretation says the jubilee system started in 1406 BC. Seder Olam's comment (that it was the 17th Jubilee), which must have been made from some other source which no longer exists, are evidence that Leviticus was written before 1406 BC, because it shows the Jubilee system had already been in operation for 17 Jubilees in the days of Ezekiel. The two independent witnesses corroborate each other.

What these independent witnesses tell us, in addition to everything else, is that the data in Kings has been preserved, there has been no errors of transmission down the centuries, at least as far as the length of the reigns of the kings is concerned. That is quite wonderful in itself.

Though the two methods give a chronology only as far back as 1446 BC, the agreement of the two methods of calculation for the date of the Exodus should boost our confidence in the whole of the Biblical record. It should boost our confidence in the accuracy of the chronology back to the time of Abraham.

I wonder how many Bible scholars down the last 20 centuries have struggled with the complexities of the length of the reigns of the kings in 1 and 2 Kings. And how many have asked "Lord, why have you allowed this to be so complicated?" Well, I think we now know - it's so there can be two independent witnesses for the date of the Exodus, providing in addition a chronological backbone/framework for the Biblical history from that time forward. I hope you understand what is being claimed and, with me, find that quite stunning: because it's quite wonderful and quite stunning.

I'm going to mention up-front that I'm a practicing Christian myself, in hopes that you might not dismiss me as a hater when I report the following: This is unlikely, as the current historical consensus is that Abraham was not a historical figure. He's more like the Romulus and Remus of the Jewish peoples.

The Abraham story cannot be definitively related to any specific time, and it is widely agreed that the patriarchal age, along with the exodus and the period of the judges, is a late literary construct that does not relate to any period in actual history

The story of Abraham, while it is indeed set much earlier, appears to itself have been written down in the Iron Age. Most credible scholars (the kind of folks who have Wikipedia pages, rather than IMDB pages) now believe it was sometime after the end of the Babylonian exile.

That being said, parts of it (particularly some of the names) appear to be very much older. So some of the story, but not all of it, is indeed very old. But there's no longer considered any reliable way to attempt to date it to the kind of precision that would allow us to talk about contemporaries of Abraham in other cultures.

More to the point, it really isn't useful to try to reason about mythic figures interacting with historical ones. The important part of the story was always what it said about the Jews as a people and how they viewed themselves (and their relationship with their God), not what it says about ancient history itself.

That AnswersInGenesis website linked in the question, if the article you linked is representative, I'd erase it from my bookmarks if I were you. ^*

One thing I found researching this is that the Archeology community has pretty much agreed that there's no good way to use their discipline to nail down a timeframe for Abraham. So when that article wants to refute something, it uses an archeologist. However, that same archeologist is nowhere to be found when it wants backup for its own hypothesis. Instead it relies on a biblical scholar who I could find no info for online, outside of Christian websites and IMDB(!). I mean he could be a totally legit researcher, but this looks super sketchy to me.

^{* - If you care about the soundness of your information, that is. I know lots of my coreligonists prefer to operate by the "unscientific method": start with a theory they want to be true, then go find info that backs it up and discard info that doesn't. Such websites are probably great for those folks, as long as you are OK with the "theory" being pushed.}