The correct answer is $\boxed{\frac{3}{16}}$.
The probability of getting a head on any given toss of a fair coin is $\frac{1}{2}$. The probability of getting a tail on any given toss of a fair coin is also $\frac{1}{2}$.
The probability of getting a head for the third time in the fifth toss is the probability of getting a tail on the first two tosses, followed by a head on the third toss, followed by a tail on the fourth toss, and finally a head on the fifth toss.
The probability of getting a tail on the first two tosses is $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
The probability of getting a head on the third toss is $\frac{1}{2}$.
The probability of getting a tail on the fourth toss is $\frac{1}{2}$.
The probability of getting a head on the fifth toss is $\frac{1}{2}$.
Therefore, the probability of getting a head for the third time in the fifth toss is $\frac{1}{4} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \boxed{\frac{3}{16}}$.
Option A is incorrect because it is the probability of getting a head on the third toss, regardless of what happens on the first two tosses.
Option B is incorrect because it is the probability of getting a tail on the third toss, regardless of what happens on the first two tosses.
Option C is incorrect because it is the probability of getting a head on the first toss, followed by a tail on the second toss, followed by a head on the third toss.
Option D is incorrect because it is the probability of getting a tail on the first toss, followed by a head on the second toss, followed by a tail on the third toss, followed by a head on the fourth toss.