The first point to note is that, from a technical point of view, VAT invoices raised by a Northern Ireland-based business can be issued in a foreign currency but you must also convert not only the value of the invoice but the VAT amount into Sterling on the invoice.
If you choose to raise an invoice this way you must convert same into Sterling by either:
A) using the UK market selling rate at the time of supply which can be found in National newspapers
B) use the period rate of exchange published by HM Revenue & Customs (also available from the National Advice Service)
Prior approval does not need to be sought from HM Revenue & Customs to use method B) above but where you have adopted same you cannot then subsequently change to another method of conversion without prior approval of your local VAT business centre.
Your Irish client may prefer to agree a price in Euro and pay you in Euro, so that they are not exposed to exchange rate fluctuations. If you want to facilitate your client, you could choose to raise your invoice in Euro with the Sterling equivalent shown on same or agree a Euro equivalent as part of a separate contract or agreement. The disadvantage of this is that you then assume the exposure to exchange rate fluctuation.
Some firms will state on their invoices that, should client companies wish to settle the invoice in Euro, they should contact their accounts department on the day of settlement to agree a suitable rate of exchange on that day.
Do not forget to consider your own circumstances and whether or not it would suit you to receive Euro at a certain point in time.