A few things to consider
1) You are unlikely to get a savings rate that will return you more interest than you pay out on your mortgage.
2) Starting to overpay by €500 now on your mortgage will save you about 9 years and the guts of €70k over the lifetime of the mortgage.
4) The earlier to start to overpay on a mortgage the better the savings. (Take a simple example, if you only ever made one overpayment and you had the option of making it in month 1 of your mortgage or month 10, over the lifetime of that mortgage, if you made no other prepayment the overall cost would differ between the two by €10)
3) And the one you are already considering - once money is paid off your mortgage you don't have access to it again so it cannot be as easily used in the event that you need it.
Unless you have a considerable cushion built up it is reasonable to me, despite the consequent cost in terms of your mortgage, to continue adding to your savings so that you have that flexibility.
I would suggest for the moment, the following approach
1) Identify a better home for your money - check the
Best Buys thread
2) Start paying €250pm into that
3) Start overpaying your mortgage by €250, this will save you about 5.5 years and more than €40,000 over the term of your mortgage (yes the maths is funny, half the overpayment but it gets you more than half of the saving)
4) Review in a year's time.
Have a play around with this
mortgage calculator - it lets you play with the numbers a bit.