Question: Why didn't Elrond send one of his sons in the company?

Answer: 
According to their [wiki page](http://lotr.wikia.com/wiki/Elrohir)

The WERE a part of the fellowship....sort of. It was their job to scout out ahead of the fellowship before they even set out. Later, they joined [The grey company](http://lotr.wikia.com/wiki/Grey_Company) , the greatest mortal soldiers of middle earth. With this great company, they apparently followed the fellowship through the paths of the dead, subsequently fighting with Aragorn and the ghost army in the battle of Pellenor Field.
Don't forget also that Gandalf states that the hope of the fellowship lies in secrecy and not in strength and that even having Glorfindel in their ranks would help.

Seeing as Legolas was already going to be travelling that direction(ish) it makes sense that he would be the one of elf kind in the fellowship