Should people who are sentenced to life in prison be allowed the death penalty as an option, and why?
Sure, why not?
If I'm understanding the question correctly, the prisoner would opt for the death penalty instead of a life behind bars, right? Sounds rather like suicide to me, and I'm generally of the persuasion that if someone wants to die it's not my place to interfere, so long as their death won't inconvenience others. So no jumping off buildings or throwing yourself under traffic. Hmmm. No blowing your brains out, either - leaving a nasty, bloody mess in your home is just inconsiderate to the poor family member who will stumble across your body. But if you have the willpower to starve yourself than by all means, go ahead. If there were legal 'suicide booths' like in Battle Angel Alita than hey, I'd say go for it.
We say that it's cruel and unusual punishment to execute someone, but when they're dead, they're dead. It's over. When they're kept in prison for life, that's psychological torture. I don't really see how it's any less cruel to deprive someone of freedom and liberties, so that they're only living a shell of a life. It's still a horrible punishment.
So if a prisoner wants to end their life I can't think of any reason to stop them. It'll save the state money. It'll reduce prison overcrowding. A potentially dangerous person is wiped from existance. *shrug* Sounds like a fine deal to me.