The politics of the matter aside, I think this would be excellent for the country for a number of reasons.
First, Obama has always been known for a profoundly thoughtful and deliberative style. This is unusual among Presidents (of any party), but is ideal for Supreme Court justices.
Second, he has a profound familiarity with the law, not just from a theoretical perspective (teaching Constitutional Law at Chicago for 12 years) but more importantly in its day-to-day practice.
Third, and perhaps most importantly, the Court desperately needs some non-judicial blood on it. Justice O'Connor, who was on the bench from 1981 to 2006, was the last justice to not have come directly from being a sitting federal judge. (She was a state judge, and before that a state senator and an assistant AG) Having a homogenous bench can create a profound myopia; people who have seen the law from a wide variety of angles will approach questions from more angles, and broad thinking makes good case law.
A former president would bring a truly extraordinary background to the job, especially when so many cases involving issues of national security, the executive branch, or the balance of powers are at stake. Far from making him biased, I suspect this would bring tremendous nuance to the conversation, with deliberations now having a profound awareness of the consequences of each decision.
The last time we had a former president on the Court was when President Taft became Chief Justice in 1921. His nine years on the bench are still remembered for the way in which he restructured the federal judiciary, making it more organized and efficient (and finally getting the Supreme Court its own building); he is still remembered as one of the greatest Chief Justices, and his bust stands in the hall of the Court. Obama, I suspect, would outdo this legacy.
There are obvious political obstacles to such a nomination, but in a day where every conceivable nomination has infinite political obstacles, this starts to fall into the category of "why not?" I certainly couldn't think of anyone more qualified for the job. (And I know some pretty exceptional jurists)
h/t +Shava Nerad