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.
The word "either" always takes a singular verb and pronoun.
"Either Mariah Carey or Lauryn Hill should win a Grammy for HER latest recording.
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
To determine whether to use "her" or "their" in this sentence, you need to consider the subject of the sentence. In this case, the subject is "Mariah Carey or Lauryn Hill," which is a compound subject.
Since there are two individuals mentioned, a plural pronoun like "their" might seem appropriate. However, when referring to multiple individuals in specific contexts such as winning a Grammy, it is common to use the pronoun "her" to refer to each individual separately. This is known as the "singular 'they'" or the "generic 'he'" rule.
Considering this, it would be more appropriate to say, "I think that either Mariah Carey or Lauryn Hill should win a Grammy for her latest recording." Here, you use "her" to refer to each individual artist separately.