I suggest a claim under the Equal Protection Clause against those states which have made voting harder for no good reason. It is not a racially based claim, nor a partisan gerrymandering claim, nor a Bush v. Gore claim. Instead I am advocating something like Burdick-Anderson balancing with teeth, where the state must prove with actual evidence that there is a problem which a strict voting law is reasonably tailored to solve.
The reason this is important is that in past rulings have not required that there actually is any fraud:
Democrats say that voter ID laws are not needed and can lead to voter suppression, while Republicans say voter ID laws are needed to prevent voter fraud, but that’s not what the courts have ruled on. They have ruled on whether the law is constitutional, in Indiana they only talked about one case of in-person voter fraud and Republicans didn’t bother trying to show any in-person voting fraud in Pennsylvania (as this link notes, the initial ruling in Pennsylvania was based on an old court case that was blatantly bigoted and endorsed the possible effect of making it more difficult for some people to vote).
If courts make it so actual fraud has to be shown to make it harder, then many of the new laws will fail since many of them actual actually make fraud easier … for voters who tend to vote Republican.