For the final return of the decedent (Form 1040) you should use her social security number (SSN). The employer identification number is used on the estate tax return (Form 1041) if there is income of $600 or more after death and before the cash and/or property is distributed to the beneficiaries.
Income that was earned while the decedent was alive would be reported on the Form 1040, while income earned after death and before distribution would be reported on the Form 1041 if applicable.