Complications are possible and delays are possible. The N-565 costs $380 in filing fees. The birth date evidence must be fraud proof to the satisfaction of the USCIS. It is possible that an old birth certificate is in the file that also confirms the improper birthdate. However, if a mistake is made, then the Alien file can confirm it. If not, those fees can be lost and perhaps more serious consequences may occur.
Sometimes, people have a good reason to change their birth date in order to qualify for a green card. In the past, people who reached 18 or 21 years of age too early could be denied a visa in soms situations. If the changed birth date would have disqualified them in the 1950s, then this can create complications or delays.
If the person is deceased, then corrections are unlikely and the application subject to denial. The applicant/certificate holder is required to make such a request, have valid identication, and provide current photographs.
I suggest that you schedule an appointment with a candid and experienced immigration attorney, rather than take any further action at this time.
The above is general information and does not create an attorney client relationship.