There are 36 possible outcomes when two 6-sided dice are rolled. Each die can land on one of 6 numbers, so there are $6^2 = 36$ possible outcomes. We want the number on one die to be a 5 and the sum to be odd. There are 3 odd possibilities for the sum: 3, 5, and 7. We want one die to land on 5, so if the sum is odd, the other die must be even. There are three possibilities when one die is a 5 and the other is even: (5,2), (5,4), and (5,6). (5,2) and (5,6) both sum to 7, while (5,4) sums to 9. So there are a total of $\boxed{\frac{1}{12}}$ ways to roll two dice so that the sum is odd and one of them is a 5.
