Jobcentre Plus staff know very little about 'signing on' (or, indeed, about benefits in general). Despite the 'Plus' bit in the name, which is meant to indicate that they provide a 'one stop' service, you can't even 'sign on' at a Jobcentre. They can't do the paperwork themselves, they've got no publicly-available computers for online claims and their publicly-accessible telephones (used by job seekers to contact employers, etc) won't accept the 0845 number used for telephone claims.
Assuming that your mother hasn't recently been in work (and paying Class 1 National Insurance contributions), she won't be eligible for 'contribution-based' JSA. (That type of JSA isn't means-tested).
She may, however, be entitled to the means-tested 'income-based' JSA if she's not got savings of over £16,000. (If she's got savings of less than £16,000, but of over £6000, any eligibility would be reduced).
You can read the eligibility criteria here:
https://www.gov.uk/jobseekers-allowance/eligibility
However the best way to check whether a claim is valid is simply to submit one. Your mother can do it online here:
https://www.dwpe-services.direct.gov.uk/portal/page/portal/jsaol/lp
She has no need to worry about being accused of trying to cheat the system. As long as they are honest, anyone can submit a 'test' claim, to see if they can get anything. (e.g. if Richard Branson was to submit a claim, and was entirely honest, he'd obviously be turned down but he couldn't be accused of fraud).
Chris