Lincoln never owned slaves. There are two likely sources of confusion that could have led to your student's remarks, though. By far the less likely is that they have confused Lincoln with his debate and 1860 election opponent, Stephen Douglas. Douglas married the daughter of a Mississippi enslaver and received a plantation as his dowry. He refused to take the whole thing but maintained a percentage interest in it until his dying day.

Abe also married into an enslaving household, which is probably what your student is thinking of. The Todds owned a slave labor camp in Kentucky which Lincoln visited at least once. (He met Mary in Illinois.) They were well-connected people so had Lincoln and Kentucky was a better Whig state than Illinois by a fair margin, so he could easily have crossed the Ohio and built a career there. It offered a clearer path to advance than Illinois, where the Whigs only have durable influence through the middle of the state and the Democrats were clearly the dominant party. An ambitious guy like Lincoln surely considered that option, but he would have had to prove himself safe for slavery if he wanted to win office in Kentucky. He would not do that, so he stuck with Illinois where he did well financially -he was a pretty rich lawyer by the 1850s- but had limited political success and few realistic options for reviving his career once it stalled out early in the decade.