The word "minute" when used to indicate "subdivisions of an angle" may appear either singular or plural. So this is not an example of a word changing meanings in the plural.
There is another alternative meaning of "minute": "(n) a brief note or summary", as in
Who is keeping minutes at today's meeting?
However, when using that meaning, the word may also appear as singular or plural.
It is possible the book is probably trying to indicate the following meaning of "minute": "(adj) very small".
This is pronounced differently, however, with stress on the second syllable instead of the first - it's a totally different word.
Compared to a the gorilla, the mouse is a minute creature.
So, sadly, I don't think "minute" is a very good example of a word changing meaning when plural as you say your book claims. (Can you tell us the book title and page?)