I am sorry to hear about your stepfather's illness. This will be a challenging time for all of you.
The Social Security Administration operates 3 programs: Social Security Retirement, Social Security Disability, and Supplemental Security Income. I will refer to them as SSR, SSD, and SSI in my comments after this. I will address each area and how it impacts your family separately.
Retirement benefits are available to eligible beneficiaries beginning at age 62 (reduced benefit), a normal or full retirement age (usually 66), and a maximum benefitat age 70. The benefit once determined remains the same (with the exception of cost of living adjustments). Thus if someone claims a benefit at age 62 it will not increase at 66 and at 70. Although you do not provide your stepfathers age, I will assume it is similar to your mothers. The SSR benefit, once applied for, begins the following month.
This effects your mother as well. Although she will need to apply, she is entitled to the greater of a benefit determined on her own earnings or 1/2 your stepfather's benefit during his lifetime. Once he passes her benefit will change to be the greater of one determined on her own earnings or 100% of his. That is the surviving spouse benefit.
No one can collect both SSR and SSD at the same time. The greater of an SSR and SSD benefit is the one that will be paid. There are also SSD benefits for dependent family members. That is why applying for SSD is important here. However, application processing through the acceptance cannot be hurried. The benefit will be retroactive to the application date.
Supplemental Security Income is a program to assist those with insufficient income. Keep this in mind in case your mother ever needs it.
You can read more about these programs at the Social Security Administration web site www.ssa.gov When the time comes, your mother should also apply for the Social Security death benefit. Please feel free to return to me any time as you go through this. I will assist in gaining understanding.