Min-entropy of a random pre-image of a random function
Since $X_v$ is a random variable which is uniformly distributed over its support $F^{-1}(v)$, its min-entropy depends only on the size of the support (and it equals the entropy). The min-entropy is at most $n/4$ if and only if $|F^{-1}(v)| \leq 2^{n/4}$. Now $|F^{-1}(v)|$ is binomially distributed $\mathrm{Bin}(2^n,2^{-n/2})$, since for each string $x \in \{0,1\}^n$, the probability that $F(x) = v$ is $2^{-n/2}$. Therefore you are interested in bounding
$$
\Pr[\mathrm{Bin}(2^n,2^{-n/2}) \leq 2^{n/4}].
$$
You can obtain a good bound on this event, which is rather unlikely, using Chernoff's bound or Hoeffding's bound. Let us know if you encounter any difficulties.