The word "either" always takes a singular verb and pronoun.
"Either Mariah Carey or Lauryn Hill should win a Grammy for HER latest recording.
I think that either Mariah Carey or Lauryn Hill should win a Grammy for (her/their) latest recording.
*My answer is their but I want a second opinion.
2 answers
Check out what happens when the conjunction is "or" not "and."
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm
and
http://grammar.ccc.commnet.edu/grammar/pronouns.htm
#3
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm
and
http://grammar.ccc.commnet.edu/grammar/pronouns.htm
#3