If the person is available for and looking for work, then they can sign on for Jobseekers payment.
Jobseeker's Benefit (JB) may be payable depending on the PRSI contribution record.
If JB isn't payable, then the person will be means-tested for Jobseeker's Allowance and entitlement will depend on any other household income, savings, etc.
There may be a period of disqualification because of leaving a job, but each case is looked at in its own right, and it may not apply if there's a good reason for it (such as you've outlined).
If the person isn't available for any type of work at the moment, the options are Illness Benefit (PRSI-based), Disability Allowance (means-tested) or Supplementary Welfare Allowance (means-tested).
Rent Supplement is subject to several qualifying criteria. An information leaflet may be found here:
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Rent Supplement is generally administered by SWA Officers who work at HSE health centres or SW offices, or in some areas, it's by postal application only.