As for individual solar systems, according to what I just looked up, stars fizzle out and become either a white dwarf, or (for massive stars) a neutron star or black hole - but not again a star in any case.
Well, that's not quite the case. As stars age, a portion of their stellar material gets dispersed in planetary nebulae. If a star becomes a supernova, he huge explosion also disperses a lot of stellar material. Even if a star collapses to a black hole, some stellar material still gets ejected via relativistic jets.
This material, which has already been part of a star, can coalesce again, creating new stars. Supernova explosions create shock waves in the interstellar gas, creating zones of high concentration, who become new star nurseries.
The first stars after the big bang were composed mainly of hydrogen and are called population III stars. During their lifetime, they created heavier elements (in astronomy-speak, "metals"). When population III stars died, they enriched the interstellar medium with those heavier elements, and the second generation of stars (population II stars) started their lives with higher metallicity. When population II stars died, the process repeated, and another generation (population I) came to live, reusing the stellar material of predecessor generations.
The Sun, for example, is a relatively recent population I star, and has comparably high metallicity.