What is the ruling on and meaning of naming a girl "Eilaf," and what is the meaning of the name "Shayma"?

The basic principle in choosing names is permissibility, unless the name involves servitude to other than Allah or is repulsive. There is no objection to the names "Elaf" and "Shaymaa". "Elaf" is derived from al-ulfah (familiarity, affection). "Shaymaa" is an ancient name, given to the Prophet's foster sister, may Allah bless him and grant him peace. It may be derived from al-sheemah (character/disposition), or al-shamah (a distinguishing mark/mole), or al-shamam (nobility/loftiness).