The answer to your question is simple:  Bud lies by telling the driver of the fancy car that Bud has run away from Grand Rapids, Michigan.  In reality, Bud has run away from Flint, Michigan.  Flint is where “the Home” is that has been Bud’s residence since his mother died.  Flint is also where Bud’s most recent foster home was located.  The foster family, the Amos family, did not treat Bud well.  Bud tells the particular lie indicated in your question because Bud wants to find the person who Bud thinks is his father.  The only clue Bud has to his father’s whereabouts is a blue flyer about a band from Grand Rapids.  Because Bud desperately wants to get to Grand Rapids, he figures lying about where he ran away from is the best way to get there. 

There are also a few more specifics about your question pertinent enough to be discussed.  The “driver of the fancy car” is actually named Lefty Lewis.  The car is specifically “fancy” because Lefty Lewis is a chauffeur and is carrying blood to a nearby hospital.

In the book Bud, Not Buddy, what lie does Bud tell the man in the fancy car?

Let me add some further specifics before answering your question.  The “fancy car” is driven by a kind man named Lefty Lewis.  The car is “fancy” because Lefty is a chauffeur who drives between one hospital and another.  In this case, Lefty is carrying blood for an important operation.  (This originally bothers Bud quite a bit.)  Further, there is a fairly simple answer to your question.  Bud lies by telling the driver of the car that Bud has run away from Grand Rapids, Michigan instead of telling the driver of the car that Bud has run away from Flint, Michigan.  The orphanage where Bud lives is located in Flint.  Further, the latest foster family (the Amos family) who Bud was forced to stay with is also located in Flint.  Bud wants to get to Grand Rapids very much because that is where he thinks is father is located.  Bud figures this lie is the best way to get him to Grand Rapids.