This is covered in the film's new Disney canon junior novelisation. In short, his reward (in the boxes we see on screen) is made up of precious metals that are evidently used in the repair of spaceships.
“We don’t keep that sort of money lying around,” Commander Willard
said. “And this is…we need every cent to keep up our operations.” “I’m
willing to accept the amount in precious metals if you don’t have the
credits on hand,” Han said, crossing his arms over his chest. “Come
on, pal, you don’t want word to get around that the Rebellion can’t
honor their debts, do you? No one would be willing to do business with
“Give him whatever he wants,” Leia said stiffly. The look she shot Han
could have incinerated half the Death Star. “The sooner he gets it,
the sooner he’s gone.”
A New Hope: The Princess, the Scoundrel, and the Farm Boy
A similar scene occurs in the earlier A New Hope: Junior Novelisation.
Stepping down to the hangar floor, Luke found Han and Chewbacca
loading small boxes onto an armored military speeder. The boxes
contained precious metals, the only form of currency that Han would
accept from the Alliance. Han had insisted on the payment, even though
the Rebels desperately needed the materials for repairing starships
and equipment. Han appeared to be completely ignoring the activity of
the Rebel flight crews and pilots.
A New Hope: Junior Novelisation
Moving down the canon scale, this is also backed up in the (alas, now non-canon) Star Wars Radio Dramatization:
Leia: Captain Solo may have no morals whatsoever, Commander, but he's right. Both Luke and I promised him payment for his and
Chewbacca's help in this matter.
Willard: Very well, I see I misjudged you, Solo
Han: I'll cry later. Right now, I'll settle for cash. In small, used notes if you've got 'em.
Willard: But this is a Rebel camp! We're hunted people, with very little Imperial currency among us!
Han: Oh. Well, your tech facilities must have stocks of precious metals.
Han: They'll do.
Willard: But those are critical matériel! We need those metals for repairs, to keep our weapons and equipment functioning!
Star Wars Radio Dramatization