Deleting a member from the records

The bishop of my ward received a letter from a member who desired to have their membership revoked and their name removed from the records. How can I do this?

I'd suggest reading General Handbook 32.14.9:

If the member still wants to resign Church membership, he or she gives the bishop a written, signed request. The bishop submits the request to the stake president through LCR. The stake president then reviews and submits the request through that system. Leaders should act on requests promptly.

It looks like it's on the bishop to process, not the clerk.

The Ward Clerk can begin the process in LCR, it's in the membership menu confidential section

I have a similar situation with this added wrinkle:

We have a person who requested to resign her membership in the Church some time ago. That was done without any difficulty. The person's husband and young daughter (under eight years old) still remained on the records of the Church.

Recently, that person who resigned her membership came to the Bishop and requested that all Ward contact with her daughter (from Primary, etc.) cease. She also indicated that she doesn't want to have to go through this again if and when they move to another location with another ward. The Bishop took this to mean that this person wants her daughter's records removed also. What is the proper way to do this?

General Handbook 32.14.9 is the section to consult.

It appears to me that a signature is required to start the process.

Note that as long as the husband is a member, they'll probably receive contact from time to time from the missionaries as a part-member family. He or she will have to communicate with the new ward to set boundaries for contact. That's not something you can put on the records.