Semi private boat tours on Naxos

I'm not sure what you mean by semi-private. I assume it's a small group of a few people, 10 or 12 at most.

Quite a few boat tour agencies in Naxos offer this type of excursion, including:

https://zastours.com/naxos?_categories=boat_tours

https://naxoscatamaran.com/

https://jasondailycruises.gr/en/

You should check with them about any access arrangements regarding your mother's mobility issues. These depend on the type of boat, its equipment, and the mandatory safety measures.

Ferry routes and schedules aren't convenient for a day trip to another island, and especially since visiting an island isn't something you can do in a few hours; once you're on the other island, you won't have the opportunity to do anything more than stay in port.

The boat tour companies I mentioned above can offer you a brief glimpse of another island, possibly for example with a picnic on a beach. You'll need to contact these companies for details and prices.

There are several boats doing day trips at the harbour at Naxos town. They all return in the early evening so you can talk to the owners and arrange a sailing.
There are also a few excursion boats at the harbour as well. They do day trips to places like Santorini, the popular Delos and Mykonos combination. You can book an excursion with any of the travel agents at the harbour.
Doing a day trip by ferry is a bit difficult because the ferries have specific routes and will not return for several hours.

Here are some images of the Delos Mykonos excursion.
Day trip Delos and Mykonos https://www.flickr.com/photos/stanbr54/albums/72157637922138466/
And for fun here are some images of Naxos.
Naxos town 2024 https://www.flickr.com/photos/stanbr54/albums/72177720318203165/