If you're using Origin 2016 you should be able to edit multiple line colors by using the "By Points"/"By Plots" tabs in the stroke color button menu. It'll look like a pen next to the fill and palette buttons.
After you choose the "Line/Border Color" tool button, and select "By Plots" tab as Ken suggested, I suggest the following steps to make the all colors distinguishable:
1) Choose a palette instead of an increment list (to avoid repeating the colors none of increment list has so many colors as 60).