At a military funeral at which both the deceased's mother and wife are present, to whom is the folded flag offered: the mother or the wife? Source? Thank you.

Usually it is going to be the wife, but a request can be put in I believe for a flag to be presented to both of them.

That's the way I understand it as well. It goes to the spouse. If a spouse is not present it may go to a parent. If both spouse and parent are there a request may be made in advance for another flag for the parent.


Mom

From the Naval CACO manual -


The Government is not responsible for conducting funeral or interment rites except in national cemeteries. Interment rites in a private cemetery are generally conducted by the funeral director and clergyman in accordance with the plans of the next of kin. If you are requested to participate in a private funeral service, you should learn from the funeral director and clergyman what is expected of you. These are certain military courtesies that you will be expected to follow. As predetermined either the Chaplain, CACO, or you will present the flag to the next of kin at the conclusion of the graveside services. If so, you may assist the pallbearers in folding the flag in cock-hat style (see fig. 1) subsequent to the ceremony. The folded flag is presented to the next of kin or his/her representative, in a dignified manner, with a little speech, such as:
"This flag is offered by a grateful Nation in memory of faithful service
performed by your loved one."
Current statutes provide that a flag of equal size is authorized for presentation to the parent if the next of kin designated is other than a parent of the deceased member. If the parents do not attend the services arranged by the primary next of kin, you will return this flag to the activity which was responsible for preparation and transportation of the remains for proper disposition or will give it to the CACO assigned to assist the parent(s) for presentation to them. It is the custom at a funeral for persons in uniform to stand at attention, remain covered, and salute whenever the body is being moved by the pallbearers and during the firing of volleys and the sounding of taps. During the prayers you should remain covered and bow your heard, but should not salute (see article 1-13, of NAVPERS 15956B). If in doubt as to your conduct during the services, you may accept the advice of the funeral director.