The earliest OED citation for "duck tape" (in the modern sense) is from the Brooklyn Daily Eagle, 21 November 1902, and it says:
"Considering... that 100,000 yards of cotton duck tape must be wrapped around the cable [of the Williamsburg bridge] with neatness and exactitude, it may be imagined that this method of cable preservation is quite expensive."
"Duck" is a strong cotton fabric which duck tape is made from; it's also used to make sails and trousers. I don't know when it became a trademark in the US. "Duct tape" came later, around the 1970s; it is of course very often used to tape up cables in ducts.