The Supreme Court has its own internal rules and does not make them public. Constitutionally there is no guidance, but I'm sure someone would look for precedent.
As such, the answer technically is that the Court itself would decide. While the Chief Justice may have more influence, the Court typically prefers to act by consensus rather than dictat, so my assumption is that they would meet in chamber and decide.
5-4 decisions are not as common as you might think- SCOTUS blog's 2013-2014 stats pack showed it was only 14% of them this year, and Anthony Kennedy was in the 5 100% of the time. As such, it would pretty much be Kennedy we are talking about, and he is still young.
If a 5-4 decision were turned into a 4-4 one, the court can choose to hold it over and rehear the case when a successor is named. Roe v. Wade, for example, was originally heard in 1971, and then re-argued in 1972, at Blackmun's request. after two new justices were sworn in. As Wikipedia states:
Following a first round of arguments, all seven Justices tentatively agreed that the law should be struck down, but for varying reasons. Burger assigned the role of writing the Court's opinion in Roe (as well as Doe) to Blackmun, who began drafting a preliminary opinion that emphasized what he saw as the Texas law's vagueness. Justices Rehnquist and Powell joined the Supreme Court too late to hear the first round of arguments. Additionally, Blackmun felt that his opinion was an inadequate reflection of his liberal colleagues' opinions. In May 1972, Blackmun proposed that the case be reargued. Justice Douglas threatened to write a dissent from the reargument order (he and the other liberal Justices were suspicious that Rehnquist and Powell would vote to uphold the statute), but was coaxed out of the action by his colleagues, and his dissent was merely mentioned in the reargument order without further statement or opinion. The case was reargued on October 11, 1972. Weddington continued to represent Roe, and Texas Assistant Attorney General Robert C. Flowers stepped in to replace Jay Floyd for Texas.
My guess, based on the collegiality of the court and it's love of consensus is that they would most likely do something like that if the issue were particularly contentious - but there is nothing that would require them to do so. It strikes me that the court could just as likely choose to honor a written opinion as a legacy to a justice who wrote it.
Finally, bear in mind that stare decisis (precedent) is important to the court, but not iron clad. As justices die, even 6-3 and 7-2 decisions can eventually be overturned. A 4-4 decision that held is thus just more likely to be challenged more quickly as things change on the ground.