I'm reading The Bully Pulpit by Doris Kearns Goodwin and she mentions that in 1889, William Howard Taft was in the running to replace an Ohio Supreme Court Justice who had died during his term of office. It appears in the book that the decision on who would be selected was in the hands of President Harrison. Wouldn't a state appointment be decided by the Governor or State representatives? Was this something unique to Ohio?
I think you have misunderstood the book.
Stanley Matthews, an associate justice of the U.S. Supreme Court, had died in 1889. Matthews happened to be from Ohio so that may be where the confusion about him being an "Ohio Supreme Court Justice" arose. But he was a (U.S.) Supreme Court justice from Ohio, not a justice of the Ohio Supreme Court.
Anyway, Taft and his patron, Ohio Governor John Foraker, lobbied President Benjamin Harrison to nominate Taft to the seat on the U.S. Supreme Court left vacant by Matthews. Instead, Taft nominated David Josiah Brewer to replace Matthews. Harrison did, however, nominate Taft for the post of Solicitor General in 1890, then to a new judgeship on the U.S. Court of Appeals for the Sixth Circuit in 1892, which Taft held until 1900.