I haven't found an answer to this:

Is there an official way to earn Han Solo? Is there a mission in which you earn Han Solo? His ally pack comes with several cards including a reward card and a side mission card which earns the reward card item. The side mission does not reward you with earning Han as an ally like Luke’s or IG-88’s after playing their side mission. The campaign mission “Fly Solo” also does not award Han as an ally if the allies win.

Trucky

The mission is Sorry About the Mess from the core set.

The green side missions grant allies, red are hero-specific (reward cards), and grey grant something else, often a reward card, but can be credits.

Han Solo 's green side mission is Sorry About the Mess, like said above, p.12 of the core campaign book.

This is also the usual way it works with allies that come as tokens in a boxed expansion - the boxed expansion comes with a green side mission, and the figure pack comes with a grey side mission - except the Rebel Saboteurs and Rebel Troopers figure packs come with additional missions to earn them (but you cannot include more than one green side mission that grants the same ally in the campaign).

@TheRealTrucky There's actually two missions you can earn him as an ally, one is Sorry About the Mess from the Core set, and the other is Hunted Down from Twin Shadows.