I don't specialize in the meanings of names but since Shekhar is the name of a Bollywood composer and there are doubts and confusions regarding what Shekhar means, I am writing this post.
The word Shekhar comes out of Sanskrit/Hindi word Shikhar which means the top, peak. Hence, Shekhar means someone who is at the top.
Going by the meaning, we see words like Chandra Shekhar, which means one who has the moon at the top of him (that is, on his head), meaning Lord Shiva.