How do I relinquish my rights to an insurance policy that was on my deceased husband?

I am sorry for your loss.  I am assuming here that the insurance policy need you as the beneficiary on the designation of beneficiary form that your husband filled out prior to his death, correct?  If you choose to renounce the policy and relinquish the proceeds you should contact the company that holds the insurance policy and tell them that you do not wish the funds be released to you.  They should have all the paperwork that you will need to fill out and I am sure have notarized.  Once you send it back it should be done with.  How the policy will then disburse depends on a few things.  Was there analternate beneficiary?  If not then the policy itself should state to whom the benefit goes in case no beneficiary is designated (and your relinquishing the proceeds in effect makes there no beneficiary). Generally then it reverts to the person's estate.  Good luck.