How do I install a GFCI in a box with two 2-wire cables and one 3-wire cable?
Easy peasy.  Most GFCI receptacles have a method to attach 2 wires to each LINE screw.
That is exactly what you do.  The two black wires go on the brass LINE screw.  The 1 white wire goes on the silver LINE screw.  And you're done.