Car Rental or Other Transportation from Dublin to Mullingar Co. Westmeath

The short answer is that a car will give you more flexibility. The long answer requires more information; specifically where do you want to go as you head slowly back to Dublin? And by slow do you mean hours or days? There is bus and bus/train service between Dublin and Mullingar Co. So public transit is an option. But whether it's convenient depends on where you are originating and where you are ending.

Thank you for reaching out with that answer. Here is some more information.
The short answer is that we will be going directly from Dublin to Mullingar Co. Westmeath on the 16th and stay in our lodgings, TBD, on the 16, 17, 18 and be done with our Convention on the 19th. (The hotel where our Convention is being held is booked so we don't know how far we will be from that venue and how we will get from one hotel to the other).
The long answer is that we return to Dublin on the 28th and we do not have any plans as yet about what we are going to be doing in the interim. We may make friends at our Convention and hang out with them during this time. Then, again, we may be shunned by everyone and be left alone to ourselves. Either way is ok with us.
Any suggestions are appreciated.

So it sounds like you have 8 days after the conference to get yourself slowly back to Dublin. Do you have a guidebook? If not, you should get one. It might help you prioritize what is of interest. Does the conference have any activities planned or suggestions for what to see/do before or after the conference? Mullingar is only 60-90 minutes by car from a number of activities: the Newgrange Passage Tomb, Trim Castle, Kells, Tullamore Dew Whisky and the entire Boyne Valley. But getting to those sites by bus could take all day. Public transit in rural Ireland is not extensive and I don't know whether rental car agencies have offices in Mullingar or if there are any organized day trips originating from Mullingar. If you are interested in private tours I recommend Beautiful Meath Tours, https://www.beautifulmeath.com/. Derek, the owner, is very nice, responsive and knowledgeable. Alternatively, you could take the bus back to Dublin and take day tours from there or bus to Galway and day tour from Galway. I'm wanting to plan my own trip to Mullingar now. Have a good trip.