Duke Street was most likely named for Brodie Duke; Duke amassed quite a bit of land in current-day Trinity Park and Old North Durham. He built his own home just to the northwest of the W. Duke and Sons tobacco factory; the Gray Map of 1881 shows Duke Street beating a straight path from Main Street to his front door, where it stopped. As the later Duke Street passed Brodie's same house to the east, the street was likely re-aligned to the east during the 1880s.
Some have speculated that the street was named after Washington Duke, as his house "Fairview" sat at the southeast corner of W. Main and N. Duke. But his house at the time of the 1881 Gray map was a small home ~ a block further east. The street may still have been named after him as a gesture of respect, but I believe geography and timing make Brodie the more likely honoree.
Duke Street continued only to Main Street originally. To the south of the railroad tracks, what would become South Duke Street was called "Lee Street." (Sometimes, this is alternatively spelled "Lea Street.") There is no clear provenance for the "Lee" name, although given the other early streets named after Confederate generals, there seems a reasonable case to speculate that Robert E. Lee provided inspiration.
See also: North Duke Street