Saint Zita the little cook was a Saint in the 1200. She worked for a very rich family who treated her very poorly. But she loved them very much. Eventually they grew to love her and became like saints just like her!
The name IS Hungarian and many girls/women bear that name in Hungary and in regions neighbouring that country. It is said that it's the result of the "Hungarization" of the Latin/Italian name Felicita(s) (English >Felicity) which means happiness and luck. The Latin appearance of the name Felicita must be at least 2000 years ago. The Hungarian form Zita might be from somewhere around 5th-8th centuries when the Magyars (now called Hungarians) began to travel closer to "Europe", or 8th-10th century when those Magyars settled down in the ex-roman province Pannonia, later to become Hungary. Because all over those periods they of course had many contacts to their immediate neighbours, the Italians/Latins/Romans.