Usually, when an author sticks himself in a book, he pretties himself up a bit, making himself just a little bit smarter and wittier and better looking.
Not Jonathan Safran Foer. His author avatar isn't exactly a flattering portrayal. Alex describes him by saying, "He did not look like anything special at all" (5.11). He's afraid of dogs and makes a big stink over having to ride in the backseat with the (mostly) harmless Sammy Davis Junior, Junior. He's a vegetarian, and his eating habits cause tons of drama in Ukraine. And even though he fancies himself a writer, he seems to be able to dish out tons of editing suggestions and criticism to Alex while not taking any himself.
All we really know is that he's looking for a woman named Augustine, who saved his grandfather Safran during World War II. And… that's it. We get some brief memories of Jonathan's grandmother, but as a character, Jonathan is probably the least interesting and least developed character of the bunch. Why do you think Foer creates a simple cardboard cut-out of himself?
Alex and Jonathan eventually stop writing to each other. Since we never see Jonathan's side of the correspondence, we have no idea why. Their only disagreements seem to be about writing, mostly because Jonathan refuses to take any of Alex's suggestions. It's difficult to tell if he ignores them because he is writing what he thinks is the truth and doesn't want to change it, or if he's just stubborn.
One way of thinking about Jonathan is like the Herschel of the 21st century. Like Alex's Grandfather and Herschel, Alex and Jonathan are Ukrainian and Jewish. If Alex and Jonathan had been alive in 1941, Alex might have been responsible for Jonathan's death, too. But instead, Alex learns about Jonathan's culture and values his friendship (which may not be reciprocated).
Jonathan might be a flat character, but maybe his story is important: maybe it can ultimately keep history from repeating its darkest moments.