Here is a typical passage from a UBF lesson (quoting from "Jesus Heals a Demon-Possessed Boy", UBF Chicago Sunday sermon 02/14/1999):
Once a renowned journalist interviewed Evangelist Dr. Billy Graham: "What would you do if you lived once more in this world?" Evangelist Dr. Billy Graham answered, "If I could live again, I would study the Bible quietly and teach the Bible person-to-person." We are doing what the famous evangelist is aspiring to do, even in his dreams.
This rumor is certainly going back to Samuel Lee who liked to make up such things. In this case Lee wanted to show how UBF 1:1 evangelization is superior to mass evangelization, and claimed even Billy Graham envied UBF for their style of evangelization. Others then simply copied Lee's sermons, without ever checking. Typical for a UBF lesson is also the vague formulation "once" and "a renowned journalist", and emphasizing the Ph.D. title of Billy Graham.
As has been mentioned before, this is far from being true. Billy Graham consistently only regretted that he did not spend more time with his family, something that is considered a cardinal sin in UBF. I quote from an interview on the occasion of his 90th birthday yesterday:
"Do you have any regrets as you look back on your life?" "I regret that I didn't spend more time with my family; I'm sure Ruth and the children paid a heavy price for all the times I was absent. I always tell younger evangelists not to feel like they have to accept every invitation they get, or be absent from home so much."
Billy Graham also added that he wished he had studied more, but I'm sure he didn't mean UBF style Bible study or even "one-to-one" with that.