Do I need a New York court to designate a Trustee for a New York Trust?

This is actually a complicated legal issue, which will be governed in large part by the terms of the trust.  When you say "New York Trust", I am not sure what you mean.  Do you mean it was part of the New York will or just made by a New York resident way back when?  Also, whether it is a testamentary trust determines if it is Surrogate's Court in NY or Supreme Court.  There is a chance Hawaii could have jurisdiction.  To answer this question I would need to see the trust.