How a neutron is able to make a nucleus stable?
A neutron doesn't make a nucleus stable.
Normal hydrogen contains only a single proton, it has no neutron, and it is stable.
Most, probably all, elements (including hydrogen) have isotopes.
These are variations of chemically the same element possessing different numbers of neutrons.
Some isotopes are stable (relatively long half life), others are less stable and can have very short half lives.
The strong force holds the nucleus together, but the coulomb force pushes it apart.
The oversimplified explanation is that the coulomb force between protons is too strong for the strong force between them to hold it together, so it needs some neutrons, which have strong force attraction, but no charge.
Stable isotopes are just that. Their half lives are (as far as we know) infinite. There are models which predict proton decay, but there is no evidence for it.
Just curious, are there any elements which have more than one stable isotope,
(given the definition of 'stable' as meaning infinite, or at least immeasurably long half life)
Yes. For examples, see any table of nuclides, e.g. this one:
For starters - H1, H2, He3, He4, Li6, Li7.
Tin has ten stable isotopes.
Separate names with a comma.