Kenmare or Dingle

The Ring of Kerry is much better for hiking than Dingle. Not only do you have Killarney National Park but also the Kerry Way, https://www.kerryway.com. The hiker in the family could combine a portion of Killarney NP with the Kerry Way by hiking in the Black Valley. As for where to stay to combine both Kerry and Dingle, consider a town north of Killarney NP. I haven't stayed in Castlemaine or Kilorglin but others here might have some ideas.

I'd pick Dingle in a heartbeat.

Kenmare is a nice enough town, and yes it's closer to that hiking. But I'd say Dingle is a lot more "special", and has at least as much (I'd say more) special stuff nearby, than Kenmare. Kenmare is fine, it's convenient, pleasant, has all the services one might need. But for me, it doesn't justify "bucket list" status. Dingle does.

On our trip (2019), we spent 3 nights in Dingle (and we would have been happy to stay another night or two, but we were on a tight schedule), and we stayed in Kenmare one night, simply because it was a convenient place to stop between the Ring of Kerry and our train back to Dublin (we did not visit Killarney National Park).

Both towns have plenty of tourists. While your family member is out hiking, I'd venture to guess that the rest of you would enjoy being in Dingle (and taking day trips out from there) than you would if you were passing the time in Kenmare. If you only have 2 nights in this area, you might want to reconsider and add another night if you can ("2 nights" somewhere just gives you one full day there). With just two nights to spend, definitely I'd pick Dingle.

Dingle. Far more interesting than Kenmare.

Have your hiking friends do the Gap of Dunloe.

I'm with David on this one.
While you can't go wrong whatever you decide, Dingle town and the surrounding peninsula would be my choice too.
That said, an excursion that you might all enjoy would be the walk thru the Gap of Dunloe. I forget whether it's technically part of the Kerry Way, but it is one of the prettiest little glens in all of Ireland. There are several ways to do it - the hiker can hike or walk it (it's 7 miles RT to the head of the glen and back), while the others could rent bikes or hire a pony trap, which can be quite entertaining if you get a particularly colorful and chatty driver.
The Alltrails website has a lot of good information on hikes in the area, complete with descriptions of the relative difficulty of each and ratings from other like-minded folks: https://www.alltrails.com/ireland
Your hiker would be well advised to bring waterproof boots if he's going to wander much off the beaten path. Even if it hasn't rained for a few days most of the trails can still have boggy sections to negotiate even in relatively dry weather.

You guys! I can't thank you enough for all your suggestions. I so appreciate it.