Best of Ireland from Belfast (7 days)

To stay within day trip distance, I think you'll mostly want to be visiting places in Northern Ireland, though you should have time for a day trip to Dublin. You could also probably fit in a trip to Bru Na Boinne which is near Drogheda, in the Republic in between Belfast and Dublin.

You can get around this area on public transport, but a car always makes things a little easier and gives you more flexibility. Some of my favourite places you can go in a day from Belfast:
-Mourne Mountains (Newcastle is the main town)
-If you like cathedrals, the one in Downpatrick is where St Patrick is (said to be) buried
-The Ards Peninisula is very nice, but I think you need car- Mount Stewart House in particular is a nice spot
-Lots of lovely places along the Causeway Coast- if you don't have a car, you could look at doing one of the tours, and visit Giant's Causeway, Carrick-a-Rede, etc
-Rathlin Island (ferry from Ballycastle)
-Derry is also doable as a day trip
-If you have a car, you can also visit County Fermanagh, the area around Lough Erne/Enniskillen is particularly nice

Within Belfast, if you like museums, definitely the Ulster Museum and the Titanic Belfast, if you like history definitely Crumlin Road Gaol, and consider a black taxi tour, and if you like shopping at all (even if you don't), St George's Market. For parks, you have the Botanical Garden next to the Ulster Museum, and low intensity hiking in Cave Hill Country Park (including a great view of the port and Samson and Goliath) and Lady Dixon Park. For a museum just out of town, Ulster Folk Park is a great place to see historic buildings.

I hope you and your mother have a wonderful time!

I'm confused about your schedule. Will you be in Ireland for a total of 7 days, including your conference? Or 7 days before/after the conference? Will you have any full days that you don't have obligations related to the conference? I'm asking because distances are short in Ireland, and while Belfast is a fine base, you might consider also spending a couple of nights in Dublin, which is only an hour away by train or bus, and has a lot to see and do. If you have 7 days in addition to the conference, I would encourage you to travel more widely.
Derry is an excellent day trip from Belfast, and I highly recommend it. Direct trains run about every hour between Belfast and Derry. The trip takes about two hours, and much of the route is along the coast, so very scenic. If you can get a seat on the right side of the train going and the left side on return, you'll get the best views. In Derry, you can walk the city walls, with or without a guide, and visit the Museum of Free Derry and the nearby Bogside Murals (again, with or without a guide).
Here is some information about train travel in Ireland:
https://www.seat61.com/train-travel-in-ireland.htm#Belfast-to-Londonderry

There is the Ulster Folk Museum, about 3 miles from Belfast and easily reached by train. Old buiildings from across Ulster have been reassembled here. There is a typical C19th town with bank, chapel, shops, inn, houses, Then there is the rural area with small farmsteads. All are furnished to represent a long gone way of life. It is a fascinating place to visit - and will probably bring back memories for your mom. There is also a transport museum just across the road.

There is the Titanic Experience, again in Belfast and beside the slipway where the ship was built and launched.

A bit further away near Omagh is the Ulster American Folk Park. This tells the story of some of the people who emigrated from Ulster to America. The park is divided into three parts. In the Ulster bit are reconstructed houses of the different families. Their histories have been researched and the costumed interpreters in the houses tell them.

You then move to a reconstruction of one of the ships the emigrees may have travelled on. Each had a small area below deck where they lived. They had to bring all food with them and had no idea how long the journey would take or exactly where they might land. It all depended on the weather.

The final section is 'in' America. Houses have either been brought from America and reassembled here or have been reconstructed. You follow the story of the different families and how they settled to life in a strange country (and strange foods - maize instead of potatoes or oats...) . Some prospered. Others did not. You could easily spend a full day here.

Wow! Thanks for the replies! I appreciate the information!