A function computable using a circuit of size $10s$ but not of size $s$
One possible approach is as follows. A counting argument shows that there are function on $n$ bits that require circuits of size $\Omega(2^n/n)$, and on the other hand a non-trivial construction that you probably saw in class shows that every function on $n$ bits can be computed using a circuit of size $O(2^n/n)$. This takes care of the case $s(n) = 2^n/{100n}$.

For general values of $s(n)$, you can use functions depending on $m \leq n$ variables, making sure that the resulting lower bound $\Omega(2^m/m)$ and upper bound $O(2^m/m)$ fit snugly around $s(n)$.