It is not correct. It should be 'Claire is one of the girls who IS on time'. Claire is the subject - singular.

Correction to my first paragraph: *(and the two subjects “Claire” and “the girls” are so close together)

Actually, this sentence is grammatically correct. Chloee's earlier interpretation is correct too: the noun phrase "the girls" is the subject of the subordinate relative clause (introduced by "who"). However, the sentence does not sound so elegant, perhaps because the subordinate clause is so short and the two subjects ("Claire" and "the girls"), or because it sounds quite close to "Claire is a girl who is on time", where the subject is Claire in both clauses. The suggested rephrasing in the earlier response (using "among") does sound more elegant/felicitous, I agree with Claire there. We can construct some sentences parallel to the original one to bring out the difference more clearly: Claire is one of the girls who roll their sleeves up and get down to work without a moment's hesitation. :) Claire is one of the girls who rolls her sleeves up and gets down to work without a moment's hesitation. :( Claire is one of the girls who have good reason to think highly of themselves. :) Claire is one of the girls who has good reason to think highly of herself. :( Claire is one of the girls who arrive on time. :) Claire is one of the girls who arrives on time. :( Claire is one of the girls who lead teams. :) Claire is one of the girls who leads teams. :( Perhaps the intuition becomes stronger if we replace the second subject with "those who" (we're just replacing a plural noun with a plural pronoun; aside from that the grammatical structure is the same): Claire is one of those who are on time. :) Claire is one of those who is on time. :(