What rights does one have regarding Next of Kin?

A person is NOT "named" next of kin. Next of kin is a legal and factual relationship. It legally means the person who is, in FACT, the most closely related, and who would have rights.
Meaning... if you are ACTUALLY the closest relative to someone.. you are that person. You cannot decline to "be" that person. For instance... if you are the child of a person, and they have no closer relatives,you cannot decide that you no longer are their child. If you are the spouse, you are the next of kin, unless you divorce.
Since you have the closest legal rights.. that would include declining to EXERCISE rights (as in the right to make medical decisions).
My children are legally my next of kin. I can choose to appoint someone as power of attorney, but that does not change that my children are actually my next of kin. I can choose to give other persons their "rights". THEY don't choose if they are/ are not my children.

Yes sure you can decline, eg if someone is ill and they named you as next of kin so you make decisions for their treatment, you can decline

Rights Of Next Of Kin

decline what? your inheritance?