Jean -
The answer will almost certainly lie in the will - ESPECIALLY if the name is a common one.
Short tutorial follows.
The people who deal with a person's estate after their death, collect in the estate3 assets, pay any debts and taxes due, and distribute the remaining estate to the people entitled to it are known as PERSONAL REPRESENTATIVES.
There are two types of personal representative: Executors, and Administrators.
Executors are personal representatives who are named in, and appointed to act by, the deceased person's will.
Administrators are personal representatives who are appointed by the Court where there is no will, or where there is a will but it either does not name any executors, or it does name executors but the named executors are all either dead, untraceable, or otherwise unable or unwilling to act.
In either case you will expect to have a Grant - that is a formal Court document recognizing the right of the Personal Representatives to act in relation to the estate and to deal with the deceased's assets. In the case of executors, this will be a Grant of Probate. In the case of Administrators, it will be a Grant of Letters of Administration (in the case of administrators appointed where there was in fact a will, it will be a Grant of Letters of Administration with Will Annexed).
If the widow was an Executor and not an Administrator, then there WAS a will (whether it has survived or not is another matter) which named her as executor. If it was a common name, then one would expect the will draftsman to have anticipated the possibility of uncertainty as to who was intended to be appointed, by saying either "my wife Martha Smith" or "my mother Martha Smith" or "My sister Martha Smith" (as the case may be) ... and this should give you the steer you need as to the executor's relationship to the deceased.
So ... the will is likely to be your key document.