Fedja is absolutely right: this has been proven, for sufficiently large $n$, by Drmota, Mauduit and Rivat.
Although it looks at first sight as though this question is as hopeless as any other famous open problem on primes, it is easy to explain why this is not the case. Of the numbers between $1$ and $N := 2^{2n}$, the proportion whose digit sum is precisely $N$ is a constant over $\sqrt{\log N}$. These numbers are therefore quite "dense", and there is a technique in prime number theory called the method of bilinear sums (or the method of Type I/II sums) which in principle allow one to seriously think about finding primes in such a set. This is what Mauduit and Rivat do.
I do not believe that their method has currently been pushed as far as (for example) showing that there are infinitely many primes with no 0 when written in base 1000000.
Let me also remark that they depend in a really weird way on some specific properties of these digit representation functions, in particular concerning the sum of the absilute values of their Fourier coefficients, which is surprisingly small. That is, it is not the case that they treat these Hamming sets as though they were "typical" sets of density $1/\sqrt{\log N}$.