Basically Claire was a girl with a very late term pregnancy when the plane crashed, she gave birth near the end of season 1. In season 2 some drug addled rock star believed he was getting visions to baptize the baby in the ocean and kidnapped him to try it, before being stopped by the rest of the people. After this Claire, who was implied to be raised in a non-religious family asked Eko, a "priest" from Nigeria about what baptism is and means and if her baby was baptized and she wasn't if that meant they couldn't be together if they didn't live, and Eko agreed to baptize them both.
The issues:
-Eko wasn't a real priest. He was a drug smuggler who obtained forged ordination papers to make it easier to smuggle heroin into the US and had been using this to live in hiding when he left Australia. Of course there was no real priest on the island.
-When asked by Claire what baptism meant, he replied that "It is said that after baptizing Jesus John the Baptist said he knew he had cleansed this man of sin." which goes against any Christian denomination's view that Jesus never sinned, perhaps to give hints that Eko wasn't a real priest.
-No Christian denomination would ever allow the baptism of an adult who did not express belief in Christ, and at no point did Eko ask Claire if she believed, nor did she ever imply it.
Of course after everything else that happened on the show later whether the baptism was valid upon leaving the island would be pretty low on the list of concerns...
I was unaware that one could be a memorialist about baptism.
See what dead0man said.
^ Bizarre BRTD hypothetical #82,093.
To be fair, this was apparently J.J. Abrams's bizarre hypothetical before it was BRTD's.
Oh, right, that makes sense considering
Lost seemed like it was written by a BRTD with even more severe ADHD.