Why can't my mom get the child tax credit for her great-granddaughter if she has full legal custody and pays for her total support?

If her only income is from Social Security she is not eligible to receive the Child Tax Credit.  She would need to have earned income from working to be eligible.  SS benefits are not earned income.

The child tax credit is based on receiving income from working.   If your mom's income comes from Social Security she is not eligible to receive the CTC.   If the grandchildren she claims were under the age of 17 at the end of 2020, she can use the recovery rebate credit to get the first two stimulus amounts for them if she did not yet receive them.

The stimulus check is an advance on a credit you can receive on your 2020 tax return.  If something went wrong or you did not get the stimulus check in 2020, you can get it when you file your 2020 return in early 2021

The Recovery Rebate Credit will be found in the FEDERAL REVIEW section. ( you should see "Let's make sure you got the right stimulus amount”)  If you are eligible it will end up on line 30 of your 2020 Form 1040.

Make sure that you enter ALL of the stimulus money received for the 1st and 2nd stimulus for yourself, your spouse and your children.

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-j-reconciling-on-your-2020-tax-return

https://ttlc.intuit.com/questions/1901539-how-do-i-preview-my-turbotax-online-return-before-filing

Click on Tax Tools on the left side of the screen. Click on Tools. Click on View Tax Summary. Click on Preview my 1040 on the left side of the screen.

And...she should be able to get the 3rd stimulus amount for herself and the children she can claim.  The 3rd stimulus is not entered anywhere on a 2020 tax return.  The IRS issues that check separately on its own timetable.

You can use the IRS Get My Payment site to see what is expected for your 3rd stimulus check:

https://www.irs.gov/coronavirus/get-my-payment

https://www.irs.gov/coronavirus/get-my-payment-frequently-asked-questions

The new and improved child tax credit, just passed in the American Rescue Plan Act, will be awarded even if the person claiming the qualifying child dependent does not work.  That credit is currently only extended for 2021, although I'm sure that the Democrats in Congress will try to extend it or make it permanent.  That's something we will all have to wait and see about.

But the current credit, including the 2020 tax year, requires that the person claiming the dependent have income earned from working.  Unearned income, such as investments, IRAs and pensions, doesn't count for credit eligibility.