What is part of speech of "fifteen minutes" in "She is fifteen minutes late"?

Solution 1:

Quite a number of adjectives such as long, high, deep, tall, late can have a complement/modifier indicating a measurement:

ten metres long, six feet tall, 15 minutes late etc.

http://www.grammar-quizzes.com/adj_modifiers.html

Solution 2:

It is an adverb phrase of time together with a determiner and functions in the like manner we are habituated to use adverbials like, Sunday morning/in the evening/at the weekend. If we shift the adverb phrase to the end we would get a sentence like this:-

• He is late 'by fifteen minutes.

To be true, minute,hour,home — though nouns — bear adverbial properties. Fifteen minutes – is an adverbial together with 'late'. 'Late', an adjective is modified by this adverbial.