Here is an observation that didn't get me very far. Let $p$ have degree $n$ and have $r$ distinct roots, all on the unit circle. Call these roots $e^{i \theta_j}$, where $\theta_k$ has multiplicity $c_j$.
Set
$$q(u) = \frac{1}{n} \sum_{j=1}^r \left( c_j \left( u + e^{i \theta_j} \right) \prod_{\substack{1 \leq k \leq r \\ k \neq j}} \left( u - e^{i \theta_k} \right) \right).$$
Then $q$ is a monic polynomial of degree $r$, all of whose roots are on the unit circle. They occur at precisely the places where $|p(e^{i \phi})|$ has a local maximum. Your goal is to show that, at one of these local maxima, we have $|p| \geq 2^{n/r}$.
I had a detailed computation of this written out, but I couldn't find a way to make it useful, so I'm recording the formula in case it helps someone else. My plan was to show that $\prod_{\{\phi : q(e^{i \phi})=0\}} |p(e^{i \phi})| \geq 2^n$, but this inequality turned out to be false; if all the $\theta$'s are very close together, then one of the local maxima of $p$ is near $2^n$ but the other $r$ local maxima can be arbitrarily small.
