Perhaps the old Daily Picayune newspaper of New Orleans, a predecessor to today's Times-Picayune, was called that because of its price. In addition to being the word for the Spanish half real piece used in the South, "picayune" was also the nickname of the US half dime.
I checked in my stack of articles from old New Orleans Picayune newspapers and found that in February 1861 the yearly subscription price amounted to five cents per issue. When sold separately the price was six cents per issue.
Interesting take on the name. I’ll bet a picayune that you are right on the money! (pun intended)