Lawyers who practice regularly before the Supreme Court are likely to prepare their arguments with a specific Justice in mind. The choice does not necessarily turn on who might be the swing vote in a given case. Often it is just a matter of which Justice can be relied upon, because of his particular interests and his insight, to search out the strengths and weaknesses of the opposing positions, and to see that all the hard questions are asked. In a labor case during the early years of the Warren Court, that would usually have meant Justice Frankfurter. Later on, depending on the circumstances, it might have been Justice Harlan or Brennan or Fortas. It has probably never been the Chief Justice.
St. Antoine, Theodore J. "Judicial Valour and the Warren Court's Labor Decisions." Mich. L. Rev. 67 (1968): 317-24.