The high court also agreed to use a lawsuit over harassment outside abortion clinics to clarify how an anti-racketeering law applies to demonstrations and other civil disobedience.
CBS News Correspondent Bob Fuss reports the Tennessee case involves an issue that comes up in a lot of states with capital punishment: whether a defendant who didn't have a good lawyer can raise questions later. There are strict time limits for appeals in many states and Tennessee says this defendant waited too long to make claims that the state didn't turn over evidence and improperly prepared witnesses.
Justices had blocked the execution of Abu-Ali Abdur-Rahman earlier this month and will now review his case. The defendant said he killed an alleged drug dealer to stop narcotics dealing to children.
One of Abdur-Rahman's government-provided trial lawyers said a defense strategy was never planned for the trial or for sentencing phase. It was the attorney's first capital case.
His lawyers relied on information they received from prosecutors that blood was found on Abdur-Rahman's clothing. The spots were paint.
Two jurors have said they would not have sentenced him to death had they known about his history of sexual and physical abuse and mental problems.
CBSNews.com Legal Analyst Andrew Cohen says the high court's decision to get involved in this case "is another sign that the justices are growing more and more concerned with death penalty procedures; that the balance that now exists in many states may not be fair enough to pass constitutional muster."
He says, "The court has gotten involved I think because of accusations that the defendant didn't get anywhere near the kind of competent representation from his attorney that the Constitution requires."
Several other big capital punishment cases are before the court, including one on whether it is wrong to execute the retarded.
In the case involving anti-abortion protests, the justices will take another look at an appeal from Operation Rescue, which was sued under federal racketeering laws. The anti-abortion group was found to have frequently violated the law in its demonstrations at abortion clinics by trespassing, blocking access and in some cases destroying property. They were fined and barred from such action in the future.
Some other groups that engage in protests, including animal rights activists, asked the Supreme Court to intervene, worried the racketeering laws could end up being used against them.
The case, which the Supreme Court will hear in the fall, raises broad free-speech questions about the treatment of political and social protests in the United States.
"Social protest has a long and revered history in this nation," lawyers for Operation Rescue wrote in court papers. "Demonstrations, even illegal ones, have been both an outlet for dissent and an instrument for social and legal change."
The high court will review whether the lower courts went too far in applying a federal law called the Racketeer Influenced Corrupt Organizations Act to the anti-abortion protests.
The Supreme Court has already ruled in the same case that the abortion clinics could sue the protesters under the law.
Now, the court will look at whether clinic blockades and violence amount to extortion under the law. It will also consider whether the law allows private groups to ask for the kind of far-reaching ban on future protests that has been been granted in this case.
"This is not a case about abortion," says Andrew Cohen. "It's a case about what is and is not, what should and should not be considered racketeering under federal law. But the fact that the Supreme Court is taking the case probably is unsettling to abortion rights groups, who because they had won at the lower court levels had wanted the high court to stay out of the dispute."
The high court is currently reviewing several death row cases with major constitutional questions. They will decide before July whether states can execute the mentally retarded and whether it is constitutional for a judge, not a jury, to decide a death sentence. They will also say when convicted killers can bring up ineffective counsel claims.
The latest case, which will be heard next fall, illustrates the court's intensifying interest in capital cases.
Justice Sandra Day O'Connor has said minimum standards may be need for capital lawyers. Justice Ruth Bader Ginsburg said among all the inmates who have asked the court for last-minute reprieves, she has never seen one who got really good legal help at trial.
Abdur-Rahman was on parole for another killing when he stabbed Patrick Daniels and his girlfriend in Nashville in 1986. The girlfriend survived.
His new lawyers want another jury to decide his fate. That jury could be told about his past. Abdur-Rahman, formerly known as James Lee Jones, was beaten by his father with a bat and a leather strap while growing up on a military base, and he was sometimes locked in a closet naked, according to Vanderbilt psychologist Linda Manning.