Has Robert Jordan or Brandon Sanderson ever said anything (or anything else in canon; I don't remember anything from the books) regarding the names of Elayne's babies? Can I name one if not?
tl;dr: As far as I know, we have no idea. They aren't born yet by the time the series finished, and no supplemental material or Word of God statements ever name them. All we know for sure is that they are not the Heroes of the Horn Calain and Shivan.
The first time this question was asked, it was RAFO'd:
HBFFERREIRA (16 AUGUST 2010)
Elayne and Rand's babies... Calian and Shivan?
BRANDON SANDERSON (17 AUGUST 2010)
RAFO ... on Elayne's babes. :)
Unfortunately, I've R'd everything that was published and never FO'd the answer, because they're never named.
There's a very popular fan theory that Elayne's babies will be the Heroes of the Horn, Calain and Shivan, because those Heroes are twins that are always born at the end of an Age. Sanderson doesn't exactly deny it when asked again:
CHRISTOPHER THAELER (23 JANUARY 2013)
Are Elayne's twins intended to be Heroes of the Horn?
BRANDON SANDERSON (23 JANUARY 2013)
It could happen.
But the Wheel of Time Companion specifically refutes this rumor. In the entry for Shivan the Hunter, it says:
He did not take part in the Last Battle as a Hero of the Horn, having been born shortly before.
Given that Elayne was still pregnant when the Heroes were called, this excludes her unborn children from being Shivan and, by extension, Calain.
Unfortunately, the article on Elayne makes no mention of her even giving birth, much less what their names were. (It basically regurgitates the events up through the end of the last book, with very little extra information.)
In Towers of Midnight, Aviendha has a vision of the future which names Elayne's Grand-daughter Talana. See the wikia. There's no info on the her children's names, however.
The only mentions of Elayne's babies in the WoT interview database at Theoryland are basically both RAFOs from Brandon Sanderson.
Firstly, from August 2010:
HBFFerreira: Elayne and Rand's babies... Calian and Shivan?
Brandon Sanderson: RAFO ... on Elayne's babes. :)
Secondly, from January 2013:
Christopher Thaeler: Are Elayne's twins intended to be Heroes of the Horn?
Brandon Sanderson: It could happen.