The time for getting the stimulus payment has essentially passed. The government is now in the process of sending out the second round of stimulus payment, and they are going to be all done sending out stimulus payments by January 15, 2021. So if your brother is not at this point eligible for a payment, there really isn't realistically anything you could possibly do that would be able to be completed by then.
However, on the 2020 tax return is a new credit called the Recovery Rebate Credit. This tax credit, which is calculated when you do your 2020 tax return this year, is a refundable tax credit (meaning you can claim it and get a "tax refund" even if you don't actually owe any taxes), and it is based on your 2020 status (unlike the stimulus checks that were sent out, which was based on your 2018 or 2019 status). You are potentially eligible for this credit if you didn't receive a stimulus payment, or your stimulus payment was too low.
In order for your brother to be eligible for this tax credit, he would need to meet the following requirements. From the IRS Recovery Rebate Credit webpage:
Generally, you are eligible to claim the Recovery Rebate Credit, if you were a U.S. citizen or U.S. resident alien in 2020, are not a dependent of another taxpayer for tax year 2020, and have a social security number valid for employment that is issued before the due date of your 2020 tax return (including extensions).
The key for your brother is that in order to be eligible for this tax credit, he cannot be your dependent. However, there is a caution here: According to the rules, it is not enough that you and he agree that you won't claim him. If he is in the situation where he meets all the requirements for being your dependent, he must check the box on his return that says "Someone can claim me as a dependent" and therefore cannot claim the credit, whether or not you actually claim him on your return.
You and he will have to look at the rules for being a Qualified Relative Dependent very closely as described in IRS Publication 501, especially the Gross Income Test and the Support Test. If either your brother's gross income is over $4200 in 2020, or if you do not pay at least half of all the money spent to support him, then he is not your dependent: He would not check the "Someone can claim me" box on his tax return and could claim the Recovery Rebate Credit. But if his 2020 income is less than $4200 and if you pay for at least half of his total support, then he is not eligible for the Recovery Rebate Credit.