Driving to west coast, in and out of Dublin, 10 days total help with itinerary please

Have you booked flights already? So much easier logistically to fly into Dublin and out of Shannon, or vice versa.

Yes. We have booked our flights already and I regret not doing that.

Louisburgh would be a good base for exploring the northern part of Connemara and Sligo - where beautiful scenery abounds ... and with a lot fewer crowds that elsewhere in the Republic.
If you're interested in a bit of Irish history I'd suggest a stop at the Doolough Famine Memorial about 15 minutes to the south. Can google it to read about what happened there and determine if it might be of interest to your group, but we thought it one of the most evocative places we visited in Ireland.
Killary Harbor is Ireland's only true fjord and there are boat trips from the town out into the estuary.
There's a very pretty loop drive around Connemara that we love, which connects Clifden with Connemara National Park (where thare are some nice hikes with pretty views), Kylemore Abbey, and the village of Cong where the "Quiet Man" was filmed back in the 50's. Nearby Ashford Castle is a favorite of many here for its Hawk Walk and for afternoon tea.
If you shifted your base to Clifden for a few days you could enjoy the Sky Road around the peninsula just west of town - it's one of Ireland's great scenic drives. Nearby is the village of Cleggan from which you can catch the short ferry ride over to the Island of Inishbofin - one of the country's better kept secrets.
There's also the option to take a day trip (or overnighter) out to the Aran Islands ... either by ferry from Rossaveel or by flying.
On the way to or from Louisburgh a good intermediate stop would be Athlone and the monastic site at Clonmacnoise.

Because you mention fun, agile and active, a great idea is to climb Croagh Patrick: https://www.ireland.com/en-us/things-to-do/attractions/croagh-patrick/. It's very close to Louisburgh. The climb can only be done by physically fit people, and at least one member of your family is 70 years old, but even someone who's that age can do it. Keep in mind that the trail is not paved and near the summit it gets somewhat dicey amid the rocks. I climbed it in 1997 in 2.5 hours. You'll be rewarded with stunning views of Clew Bay and the Atlantic Ocean if it's not foggy.

I also recommend a visit to the Ceide Fields (https://heritageireland.ie/places-to-visit/ceide-fields-neolithic-site-visitor-centre/) in north County Mayo. It's an archeological site dating to 3,500 B.C. And the views from the cliffs are the best I've seen in five trips to Ireland.

One more idea? A visit to the Aran Islands (https://www.aranislandferries.com/) is worth it.

You’ve had lots of good advice above and could easily spend the six days in and around County Galway and Sligo, especially if you plan for an overnight stay on the Aran Islands. You could do this from one location, or else split between two - Louisburgh/Wesport and Galway city. (This would also allow a day to explore the Burren to the south with its limestone scenery which is completly different to anywhere else in Ireland. )

Somewhere else I would add to the list is the National Museum of Ireland at Turlough Park which is an excellent folk museum. As well as loads of artefacts about the way of life over the last 100 years it has a lot of videos made in the 1950s and 1960s of people talking about their way of life and customs. There is an interesting short promotional video about the museum here. We spent several hours here and could have easily spent a full day here.

These are wonderful suggestions. Thank you.
I am wondering it we have time to see the areas around Killarney on the way to or from Dublin and then up to Galway and Louisburgh or the other way around as the itinerary is wide open. This is likely the only time my mom will ever be to Ireland and I want to show her the best that I can and we will have a car. TY

Trying to include Killarney will be a challenge, but would be doable if you're willing to spend more time in the car getting from place to place.
One option would be to drive from Dublin to the Killarney area and spend a couple of nights seeing the National Park, the northern half of the Ring of Kerry (including the very pretty Skellig Ring Road), and some of Dingle. From there you could head up the coast along the N67 which is a very pretty way to go. That would include a short ferry crossing of the Shannon at Killimer, and could include a stop in the cute little beach town of Kilkee for lunch or coffee ... with maybe a walk along the nearby cliffs which you'll probably have all to yourselves. Lots of great photo ops there, and a continuation of the route north would permit a stop at the Burren as was mentioned.
Thence north to Louisburgh and Connemara.
Another possible option would be to continue your journey on your arrival day (since you'll already be at Dublin airport anyway) and hop on one of the twice daily flights via Ryan Air to the Kerry airport near Killarney and collect your rental car there for a one-way journey back around to Dublin. Would save you a half day's drive across the island.

I agree with Robert that there is so much to see and do near Louisburgh. IMO, you'll show your mother the best of Ireland by focussing on the area where her family is from, rather than shoehorning it into an otherwise by-the-Steves-guide trip. In five trips to Ireland, I have had the best experiences in the places that have a family connection for me or my travel partners. Even on my last trip when my friend with zero Irish ancestry wanted to kiss the g.d. Blarney Stone because her father and grandmother had done it. Before your trip, find out as much as you can about the family sites, even if it's only where someone thinks the house used to be. Visit the graves. There is enormous power in walking those lands. You will not regret skipping Killarney (or leaving to next time). I promise.

All the guide books plug places like Killarney and Dingle and everyone seems to think they HAVE to do them... The scenery in Connemara, Galway and Sligo is just as good and there won't be as many visitors either! The Sky Road Drive from Clifden on a sunny day must rank as one of the best short drives in the world. The tiny village of Roundstone with its harbour is idyllic and the coast round there is special too. There is the romantic Kylemore Abbey and gardens or a boat trip on Killary Harbour (described as Ireland's only fjord.)

Ireland is the place to slow down and enjoy. Too many people try and push in everything. Your mom won't be short changed iif you don't include Killarney!

Adding to the excellent advice above for Connemara and Croagh Patrick--we enjoyed seeing a sheep dog demonstration at Killary Sheep Farm overlooking the fjord. Also a day trip to Inishbofin (thank you Robert for that suggestion). I agree with skipping Killarney this trip and enjoying Connemara and County Mayo. You might have time to go south into County Clare and see the Burren.