If my half sister hired a lawyer and filed a petition to administor our father’s estate which is intestate, how do I go about being named administrator instead?

I am so sorry for your loss and for the issues that have arisen.  If you contested the appointment with in the time constraints permitted under the law (generally 30 days in most states and in writing) then you have followed the necessary procedure for the prompting the next step: waiting for the court to hold a hearing on the matter.  Many states have statutes that list the order of priority for appointment of Administrators in intestate estates.  generally speaking a spouse has first priority followed by children.  Your half sister is still your Father's child and she has a right to apply.  You must also have grounds and "good cause" to object.  You cannot simply object because you want the job yourself. If that is the case you should have filed your own petition with the court. To contest the appointment, you must have a statutory reason for his challenge under your state’s laws. For example, in New York, you might allege that the petitioner is incompetent, doesn’t live in the state, is a convicted felon, or prone to substance abuse, dishonesty, or rash decisions. But you would have to prove these allegations so you can not make them willy nilly. The court will ultimately decide and you may both be appointed.  Best to try and get along about things now.  Otherwise it will be a long and bumpy road to completion. Good luck.