In his paper "On the size of Kakeya sets in finite fields", Dvir introduces the notion of a $(\delta,\gamma)$-Kakeya set, which is a set $K\subset F^n$ such that there exists $L\subset F^n$ of size $|L|\ge\delta q^n$ such that for every $x\in L$ there is a line in the direction $x$ intersecting $K$ in at least $\gamma q$ points. He then proves the following result:
Theorem. If $K\subset F^n$ is a $(\gamma,\delta)$-Kakeya set, then $$ |K| \ge \binom{n+d-1}{n-1}, $$ where $d=\lfloor q\cdot\min\{\delta,\gamma\}\rfloor-2$.
To what extent does this answer the question?