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

Answer: 
To supplement, rather than supplant, Jason Baker's excellent answer above, I can only offer the following note. In a sense, Elrond *did* send his son on the quest. In *The Tale of Aragorn and Arwen*, which is found in the Appendices to *The Lord of the Rings*, we read this passage:

> 
> [After his father Arathorn's death] Aragorn, being now the Heir of Isildur, was taken with his mother to dwell in the house of Elrond; and Elrond took the place of his father and came to love him as a son of his own.
> 
> 
> 

There are several other references to the relationship between Elrond and Aragorn, and almost all of them paint the relationship as one between a father and his son, including this one, from the same source:

> 
> "**My son**, years come when hope will fade, and beyond them little is clear to me. And now a shadow lies between us. Maybe, it has been appointed so, that by my loss the kingship of Men may be restored. Therefore, though **I love you**, I say to you: Arwen Undómiel shall not diminish her life's grace for less cause. She shall not be the bride of any Man less than the King of both Gondor and Arnor. To me then even our victory can bring only sorrow and parting - but to you hope of joy for a while. For a while. **Alas, my son!** I fear that to Arwen the Doom of Men may seem hard at the ending." 
> 
> 
> 

Note that here, Elrond doesn't say that he loves Aragorn *like* a son; he literally calls him "My son". 

Of course, Aragorn did indeed go with the Fellowship of the Ring, so one of Elrond's sons was among the Fellowship.

So, as Obi Wan Kenobi would say, Elrond did send his son to accompany the Ring on its journey to Mordor - "from a certain point of view".
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