Mayo is a large county. That's even by Irish standards.
Comparing it to places in England, it's a bit bigger than Norfolk and has a population that is lower than any London Borough. So bear this in mind when you are thinking about what to do. There's countryside, there's coast, there are historical sites, there's more countryside and more coast. I don't know what I would especially take a 3 year old to.
Belmullet is peripheral. Find out what the relatives do!
If it was me and I didn't know anywhere I'd check out:
Nephin National Park (there's a visitor centre and a lot of park)
Ballina. Spot the Welcome Joe Biden signs and US flags that are still very much up. There's a Dunnes one side of the car park and a Lidl the other.
Castlebar. I like it as town. National Museum - Country Life site very close. If you are into 1798 history (and what 3-year old would be) it is on a Humbert trail. Remember to start at Killala.
Surviving Big Houses etc. that survived, contain something to see and have grounds: Westport, Belleek, Enniscoe (Heritage Centre there).
Ceide Fields. I can get mildly exited about field systems but have a late Primary aged child who has learned to humour me.
Beaches. Too many to choose from, but go when it's not raining.
I love Mayo. I'd be going for me though.