The name Lainie is a girl's name meaning "bright shining one". Lainie and is often added to lists like Inspirational Baby Names That Mean Hope and discussed in our forums with posts like "Pick 5 (GIRLS)".
From the experts:
Lainie is a nickname for Elaine that has been in the public eye via Lainie Kazan (that was her full given name), who was a popular singer in the 1960s and 1970s (she was Barbra Streisand's understudy in "Funny Girl"), and then became a motherly character actress. We still find Lainie a bit flimsy for a full first name.