What are his interests? It will help you get better answers. Just to throw some things out there, I'll assume he is an active/sporty type who probably prefers to spend his time outside. Depending on what your travel dates are, you may want to look into seeing if there are any GAA hurling or football matches you can get tickets for, or check with some of the local/county clubs to see if you can catch a game (you don't mention what cities and towns you are thinking about overnighting in). There is good surfing in Ireland (Donegal, Clare, Dingle, etc) and if he knows how to surf (or wants to learn how to surf - there are schools around the country) that might be fun (he'll need to rent a wetsuit, even in the summer the water is pretty cold). If you want to drop some bucks and stay in a castle overnight, Ashford Castle offers all sorts of outdoor activities like archery, fishing, horseback riding and even falconry lessons. Plenty of hiking opportunities - you could tackle Croagh Patrick. An overnight on the Aran Islands would be fun - you can fly or else catch the ferry over (although I will mention that the ferry from Rossaveal is far more reliable as the others can be cancelled if conditions are not ideal - even on sunny days, if the sea is too choppy, it will be cancelled). If you go to the main island (Inis Mor), he can hike up to Dun Aengus. You can also hire a pony & trap to catch a ride around Inis Mor. Another idea for a day trip by boat could be if you go to Skellig Michael. Ailwee Caves might be worth a quick stop if you head over towards the Burren in County Clare, as is Poulnabrone Dolmen also in the Burren.
In Dublin, people might recommend the Guinness Storehouse, but again, he can't drink at the end of the tour up in the Gravity Bar (he can enjoy a marginally ok view) and most of the museum is dedicated to Guinness advertisements - so don't expect to see the full brewing process - you'll hear about it, read about it, but won't see it - and if all he wants is a tee or hat with the logo, he can buy that at just about any gift shop in Ireland (including the airport).
He can get into pubs if you want to listen to some music in the afternoon, but some pubs will not let him in during the evening. In Dublin there are some museums and churches that might hold a bit of interest for a 16 year old boy. The National Museum has some fine examples of Irish bog bodies, and the crypt underneath St. Michan's Church has some mummified bodies laying around in open coffins. Kilmainham Gaol is also a great place to visit in Dublin.
If you make it up to Northern Ireland and hit the Antrim Coast, there is Carrick-a-Rede rope bridge which is seasonal. And of course, the Giant's Causeway and Dunluce Castle. In Belfast or Derry, you'll see some fantastic murals on the streets and you can do a Black Taxi tour in Belfast or there is the Museum of Free Derry if he wants to learn about The Troubles, Bloody Sunday and the Irish Hunger Strike.