I am the executor of my Father-in-Law's estate. Prior to his death last January, he had about $5000 income. After his death, his estate income was less than $600. (He had a single NOK/beneficiary, so the assets were quickly distributed.) I know I need to do a 1040 for his income prior to his death. Do I need to do a separate 1041 for the less than $600 after his death, or can I just include that in his regular return, or should I just ignore it (since it is below the threshold)?
".... should I just ignore it (since it is below the threshold)?"
You can simply ignore it since the income is indeed below the threshold and you do not need to include the small amount of income that was attributable to the estate as income on another return (the $600 estate exemption would cover that).
See https://www.irs.gov/instructions/i1041/ch01.html#d0e340
However, if you obtained an EIN, it would be advisable to file a return since the IRS will almost certainly be expecting one and, by filing the return as the initial and final return, the return will officially result in the termination of the estate as far as the IRS is concerned.