This is a hotly debated subject (obviously), so all of the reasons I list here are to be judged of your own accord. On eNotes, we are not supposed to answer your homework questions for you, so rather than giving a list of ten reasons, I will instead offer some of the common arguments, from which you could draw many reasons from.
Some argue that prisoners have the right to vote from the Constitution. Stripping away certain populations' ability to vote could be seen as going against the idea of self-governance that our country is built upon. Additionally, our prison system is known by some to discriminate against people based on their identity. It is quite clear that denying someone the right to vote based upon aspects of their identity is wrong, so you could argue that denying prisoners this right is legally wrong because of this.
Another commonly made argument surrounds the fact that an incredibly large number of Americans are incarcerated. To ignore their vote as a whole is ignoring a relatively large portion of our society. Additionally, some say that it could help with prison overpopulation to allow prisoners to have their voices heard, as they could potentially help get rid of some of the laws that are wrongfully imprisoning people.
There are a few other legal arguments to be made. Prisoners are afforded free speech, and getting rid of their right to vote can be seen as infringing on these rights. You could also say that, for those states where ex-felons cannot vote, that this is an infringement of this right even after they are out of prison.
Try to look at the issue from a legal perspective, as well as a moral one, and plenty of reasons to support inmates right to vote will appear. Good luck!