He didn't want to offer his own name. There was a conflict of ego competing with self preservation in his hesitation to respond to her. If the people of the planet were supporters of the Resistance he couldn't afford to let them know they had Kylo Ren in their grasp. But if it turned out the planet was neutral or under First Order control, he would potentially find allies or at least some fearful respect.

He wouldn't offer his name willingly to anyone, but for the moment he decided to give her his.

"Kylo Ren." His response comes delayed, after they start walking. But at least he offers it. A small gesture of compromise for her going ahead of him.

"Have you seen anyone that does not look like a half drowned wet dog?" Such flattery he offers. Truly Ren knows how to win a woman over with words and compliments.