Can a supernova make a new star?

This idea has been around for decades, so I'm not sure who first came up with it. Here's a reasonably sourced article on the involvement of supernova in solar system formation:Exploding Star May Have Sparked Formation of Our Solar System

The shock wave from an exploding star likely helped trigger the formation of our solar system, according to a new 3D computer model, researchers say.
The solar system is thought to have coalesced from a giant rotating cloud of gas and dust known as the solar nebula about 4.6 billion years ago. For decades, scientists have suspected a star explosion called a supernova helped trigger our solar system's formation. In particular, the shock wave from the explosion is thought to have compressed parts of the nebula, causing these regions to collapse.
According to this theory, the shock wave would have injected material from the exploding star into the solar nebula.

So yes, material from supernova can end up triggering the formation of new suns, and material from supernova do end up mixed in with the nebular material that makes the new stars.