I agree with Ben Barber that you should not try to choose a PhD problem by picking from a list on MO, or anywhere else on the public Internet. However, there is one useful non-obvious piece of information we can give you:
There are a lot of mathematicians doing research that heavily involves Galois theory, however, very few of them would call their research area Galois theory.
In fact, almost every subfield of algebraic number theory heavily involves Galois theory in one way or another. Professors who describe their research area as "algebraic number theory", "Galois representations", "Iwasawa theory", or "the Langlands program", to name a few possibilities, could most likely give you a problem heavily involving Galois theory if you were their PhD student.
You could therefore apply to a PhD position at schools strong in number theory. You should speak to professors at your own university, in particular ones who might be writing you recommendation letters, to decide which schools in particular you should apply to.