1. This picture
Ehrenreich pretty much has the vibe and the look right here. Jaded, beaten down by the world, but also stern and ready for a fight at any moment. I wouldn't be surprised if his people chose this as his main photo on IMDB because it feels so right for Han Solo. 

2. He's pretty much already done a young Han Solo movie
"Beautiful Creatures" is a young adult movie about witches, in which Alden plays a cynical teenager who stumbles into that secret magical world and masks his earnestness with disarming humor before turning into a sincere idealist by the end. That's pretty much exactly Han Solo. And Alden is great in "Beautiful Creatures," by the way.

3. His oblivious and sincere performance in "Hail Caesar!"
Alden's Hobie Doyle is hardly a Han Solo analogue in the Coen Brothers' comedy but Hobie does go through an evolution that is similar to that which we have to imagine a young Han Solo movie could follow. Specifically, he is a kid who came from nothing to become a movie star, and he's enjoying the good life until he gets caught up in studio machinations and the weird conspiracy at the heart of the movie. 

4. He's got the comedy chops
Young Han Solo has to be funny, but in a wry kind of way. "Beautiful Creatures" shows that off directly, but his "Hail Caesar!" performance demonstrates that ability differently because it's all deadpan delivery -- like when he glances directly into the camera in this shot.

Directors Phil Lord and Chris Miller
Anyone this pair would have chosen would be perfect. Just in case you had some deeper fear, you shouldn't worry. Lord/Miller can do no wrong. They could cast, say, Ted Cruz as young Han Solo and it would still probably be a pretty good movie. Remember when everybody thought their "Jump Street" (pictured) movie with Jonah Hill and Channing Tatum was a terrible idea?