As well as having 13 paid PRSI contributions before being eligible, you'll need to meet the other PRSI qualification rule for Illness Benefit, which includes the following (taken from the page linked earlier):
39 weeks of PRSI contributions paid or credited in the relevant tax year, of which 13 must be paid contributions.
If you do not have 13 paid contributions in the relevant tax year, then 13 paid contributions in one of the following tax years can be used instead:
Either of the two tax years before the relevant tax year
The last complete tax year (before the year in which your claim for Illness Benefit begins)
The current tax year
or
26 weeks of PRSI contributions paid in the relevant tax year, and 26 weeks of PRSI contributions paid in the tax year immediately before the relevant tax year.
The relevant tax year is the second last complete tax year before the year in which your claim for Illness Benefit begins.
If, for example, you applied for illness benefit in 2015 (after you've been in work for 13 weeks), SW will look at the current tax year (2015), the last complete tax year (2014), the relevant tax year (2013) or either of the two years before the relevant tax year (2011/2012) to find 13 paid contributions. You will have been given credited contributions for your time on Maternity and Illness Benefit. If you've been out of work for 3 years, you may have no paid contributions for 2012/13/14. If your employer paid you while on Maternity leave, you might have some paid contributions.