Luke hit the small button for the sub-light engines, jumped out of hyperspace, and immediately hit the scanners. There were no ships, no life, no sign of anything nearby. Another dead end. This was his tenth stop along the last route where the Millennium Falcon fell out of hyperspace. He checked his clocks again and realized the hour was late, and only getting later. He was overdue in getting to the rendezvous point. He knew they probably would not wait for him, and he would be assumed to be dead. That is what they were told, and he was running out of time.
He also knew he should leave now and give up the search. There was every reason to believe Han was fine. The ship had dropped out of lightspeed, but there was no reason to believe that the ship could not maneuver. It is also possible he ended up too close to a star and then it would be too late to help him anyway. Besides, he really barely knew Han, he just met him only days before, and obviously he has had a questionable past.
Still, there was something that drew Luke to Han. A feeling of brotherhood, fighting in the Death Star together and the escape as well. He trusted in Luke to man the guns of the Falcon. It was Luke's first taste of that sort of combat and he would never forget the camaraderie he felt with Han. That, and he definitely owed Han. If he had not come back Luke would have never made it that extra stretch down the trench along the Death Star's surface. He would have been destroyed like all of the rest, and the Rebellion would be over.
Luke also had a gut feeling that Han needed help. It was something he couldn't explain, but recent events have caused him to trust that gut instinct much more than he used to. So he switched on his navigation controls and plugged in the exact same jump trajectory. He would continue to look for Han until his fuel was spent. It was the least he could do for his new friend.