What makes a monatomic ion stable?

Answer 1

There is a certain stability within an atom when the electrons have a full valence shell (8 electrons in the outer layer). This stability is caused by the interaction of all the sub-atomic forces but the bottom line is that when there are 8 electrons in the outer layer, the atom is in a very stable configuration.

When an atom has either 1 or 2 electrons off from this stable configuration (either having extras beyond the full layer or having empty positions in its outer layer) then either giving up electrons (making it a cation) or taking in electrons (making it an anion), provides this stable configuration. That's what makes a monatomic ion stable (the full valence shell)..