How can a sibling start the process of probate?

I am so sorry for your loss.  I do not think that you have a correct understanding of the statute in North Carolina because the statute that I have found states as follows: "(b)        Letters of Administration.  Letters of administration shall be granted to persons who are qualified to serve, in the following order, unless the clerk of superior court in his discretion determines that the best interests of the estate otherwise require: (1)        The surviving spouse of the decedent; (2)        Any devisee of the testator; (3)        Any heir of the decedent; (3a)      Any next of kin, with a person who is of a closer kinship as computed pursuant to G.S. 104A‑1 having priority; (4)        Any creditor to whom the decedent became obligated prior to his death; (5)        Any person of good character residing in the county who applies therefor; and (6)        Any other person of good character not disqualified under G.S. 28A‑4‑2." Additionally, the fact that they are out of state would probably require that another party in state is also appointed, and that would be one of you.  If you think that I am incorrect here please check with an attorney in your area. And consider having them renounce their "right" as you put it to be appointed opening the door for you.  Good luck.