tax break for people on social security income when renting an apartment.?

This isn't a part of Social Security but rather Fair Housing. Google "Section 8 Housing" to see the requirements. If you Mom has low income and virtually no assets or savings she may indeed qualify for a Section 8 voucher to help with her housing costs.

None of what you said is true. Some cities have a Dept of Health Services which pay part of the rent. Other people have a Section 8 voucher which pays most of the rent. Being on SSI by itself does not help pay the rent. How can you mother have a tax break if she does not work and file a tax return. That is ridiculous. There is no such "tax break".

Please put the dollar sign before the amount not after. It is disconcerting and incorrect.

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her SSI is based on her needs

if she finds an apartment for less, her needs are less and SSI will be less

the landlord is not giving her a break because she is on SSI

she might be eligible for Section 8 housing assistance

Not just for being on SSI, but she might qualify for a Section 8 housing subsidy.