Expected number of maximal cliques in $G(n,p)$
Since $\mu$ is the sum of $n$ terms, we can estimate it up to a factor of $n$ by the maximal term:
$$ \max_{d=1}^n \binom{n}{d} p^{\binom{d}{2}} (1-p^d)^{n-d} \leq \sum_{d=1}^n \binom{n}{d} p^{\binom{d}{2}} (1-p^d)^{n-d} \leq n\max_{d=1}^n \binom{n}{d} p^{\binom{d}{2}} (1-p^d)^{n-d}. $$
In order to find out which term is the largest, we can use the estimate
$$ \log_2 \binom{n}{d} p^{\binom{d}{2}} (1-p^d)^{n-d} \approx nh(d/n) + \binom{d}{2} \log_2 p - (n-d)p^d, $$
which is true up to polynomial factors for $d = \omega(1)$. We can estimate the magnitude of the largest term by differentiating this quantity with respect to $d$. If you follow this path, you should obtain an estimate for $\mu$ which is tight up to polynomial factors in $n$.