From your question:
"I was gifted $100,000 in 2017 from my father. We were told due to his lack of income . . . "

While there was an obligation to file the Form 709 Gift Return prior to the filing of your father's Form 1040 for 2017, it sounds as if your father, whether individually or jointly, has possession of an estate that is less than $5.6 million (2018 exemption) in total value, at this time.

N.B.: 2017 amounts were
a) Lifetime Exemption $5.45 million and 
b) annual single person exclusion @ $14,000.

As was stated in TurboTaxMichaelL1's answer, but to clarify this point, the usual Form 709 filing simply reports that of the entire Lifetime Exemption (the $5.6 million in 2018 - it was minimally smaller in 2017) the Gift amount is to be accounted for against that.  However, this is further reduced by the fact that of the $100,000 Gift, your father could and can legitimately characterize not less than $15,000 as an excluded gift.  Gift exclusion in 2018:  One person can give another person (not a referring to a spouse) as much as $15,000 in one year with the Gift being excluded from the obligation to report on Form 709.  If a pair of parents Gifts, then it would be twice 2*$15,000; or if a parent gifts to a married child (the child and spouse), it would be twice.  Thus, it is possible that not less than $15,000 would be excluded and if both parents were surviving and the recipient was married, that would exclude $60,000.

Lastly, as Michael said, no Gift Tax is due, if all gifts made during the Lifetime, total less than the Lifetime Exemption, so no penalty is assessable and there is no practical need for a Form 709.  However, if the gift is to be repeated, good practice suggests that retroactively for the prior gift, and going forward for future non-excludable gifts, Forms 709 be submitted.

Yes, he can file the gift tax return late and should file it.

If he uses up some of the unified credit the tax on the gift tax return will be $0.

There would be no penalty due to no tax.

You might want to seek a professional to file the gift tax return and make sure a copy is kept of it.

If your filing is more than 60 days late (including an extension), you'll face a minimum additional tax of at least $205 or 100 percent of the tax due, whichever is less. Failure to pay the gift tax on time will result in a penalty of 0.5 percent month of the amount due, up to a total of 25 percent. ---Which would be $0 with no tax due.

I am assuming that this is for the tax year of 2023.According to IRS regulations, the only exclusion available for not having to file the gift tax return is if the the gift is 17,000 or less. See HERE pg. 2 and 3 and HERE under "Do You need to file a gift tax return." No where is it mentioned that there is any exclusion based on the taxpayer's lack of income or income strictly of social security. 

In addition, there is a lifetime exclusion of 12.92 million per donor spouse which cover any gift tax that would be due above the 17,000 threshold. See HERE under "How Much can you give tax free".

Finally, the deadline for filing the 709 gift tax return for 2023 is April 15,2024. If April 15 falls on a Saturday, Sunday or

legal holiday, then the form would be due on the next business day. Late payment penalty is 1/2 of 1% of any tax not

paid by the regular due date. It is charged each month or part of the month the tax is unpaid  with a maximum penalty of 25%. Late filing penalty is 5% of the tax not paid by the original due date for each month or part of the month that your tax return is late with a maximum  penalty of 25%. Unless, you maxed out your lifetime exclusion, you would most likely not have to be too concerned about these penalties. See HERE  pg. 2 for more detail.