Lietuva is a Lithuanian word for 'Lithuania'. It is believed that originaly it was a name for one small Baltic tribe that lived between rivers Nemunas and Neris (somewhere around the area where the city of Trakai stands nowadays).
The word Lietuva itself originates from a small creek Lietava or Lietauka (this creek flows into river Neris). The Lithuanian word 'lietus' means 'rain', so the creek might have got its name from there.
Lithuania was first mentioned in written sources in 1009AD (chronicles of Kwentinburg) when Lithuanians have brutally killed bishop and missionary Brunon who has come to Lithuania to bring christianity (not a terribly good way to make a first impression to the world). I might add that Lithuania was in fact the last country in Europe to accept christianity somewhere in the late XIVth century.