No evidence but tradition, but why would there be? The most bold claims about Abraham's actions include him naming the city of Beersheba (which still exists, but could have been named any time in the past forty centuries), that he raised an "army" of 318 men to chase Mesopotamian influence out of Canaan (at that scale his army wouldn't exactly leave an impression), and the Muslim tradition that he and Ishmael built the Kaaba (Mecca is quite the round trip from Palestine for a father/son construction project). Outside of that, there's little to separate Abram/Abraham from any other somewhat-wealthy Cannan shepherd.

There's no evidence he existed, but no evidence doesn't necessarily = proof he didn't exist. There's not a lot of reason to think he did.

His supposed tomb still exists and is one of the more contentious sites in the Holy Land.

We wouldn't really expect to see any evidence of an individual Chaldean shepherd in Canaan over 4,000 years ago.