What is the minimum number of selections from the set { 1, 3, 5, 7, 9, 11, 13, 15 } that we need to guarantee that at least one pair of those numbers adds up to 16?

Perhaps I don't understand the question. You only need to choose one pair that adds to 16, any of {1, 15}, {3, 13}, {5, 11}, {7, 9}.

You can choose up to 4 numbers without getting a pair that adds to 16, so if you choose any 5 numbers, at least one pair of them will add to 16.