The Sun is actually a THIRD generation star. What I mean by this is that there are chemical elements in the Sun that were made inside another star, but that star itself can only have made those elements because it had material in it that must also have been made inside previous, second generation stars. Eventually we get back to the first generation stars, born out of primordial gas from the big bang that contained almost no heavy elements (those beyond helium) at all.
That is quite a mouthful, so let me explain using an example - barium.
There is barium in the Sun. We can tell that by looking at the spectrum and seeing absorption lines due to barium. But barium cannot be made in the Sun. The barium is made via the s-process, which involves the slow capture of neutrons onto the nuclei of iron-peak elements. This happens during the asymptotic red giant branch phase of stellar evolution, and the Sun has 6 billion years or so before it gets to that point. [NB: not even half the abundance of chemical elements beyond iron are produced by supernovae explosions$^1$.]
So, before the Sun, there must have existed a star - probably an intermediate mass star - which evolved to become a giant, made barium in its interior, then lost its envelope through a massive wind into the interstellar medium, and that material was incorporated into the protosun. Such stars (between, say, 2 and 10 solar masses) would have much shorter lifetimes than the Sun$^2$, so plenty of time for them to live and die before the Sun was born.
But wait a minute! That previous star must have already had iron-peak elements in its interior to act as a "seed" for the s-process production of barium. These were not and could not be made inside that star. They must have been made in a previous star, probably a massive star, that burned through all the nuclear fusion stages before exploding as a supernova, casting heavy elements, including iron-peak elements, into the interstellar medium. This previous star could also have had its own (metal-rich) ancestors, but ultimately as we go back in time we reach a point where the previous star was a first generation star, made from primordial H/He gas, with almost no heavy elements. These first generation (a.k.a. population III stars, just to be confusing) were probably very massive and short lived - a few million years. They would be born when the universe was a few hundred million years old and we can see no examples of them in our Galaxy today.
To try and define more precisely what I mean by "generation".
- First generation - made from primordial big bang material.
- Second generation - a star made only from the detritus of dying first generation stars, enriched in heavy elements but lacking in primary s-process elements.
- Third generation - a star made from material already enriched in heavy elements and including elements that are produced in the s-process inside previous second (or third) generation stars.
So that is why I claim the Sun can be classed as a "third generation star" - it contains atoms/nuclei that must have been inside at least two previous stars.
But you should not take this too literally. There are grains of material trapped inside meteorites that consist of solids that were already present in the pre-solar material. These are important because these grains were thought to have formed in individual stellar events and their isotopic compositions can be studied. These tell us that the Sun formed from material that has been inside many different stars of different types.
Stellar evolution and nucleosynthesis calculations tell us the same story. For example, whilst most of our oxygen was made in massive stars that underwent a core collapse supernova, such events do not produce that much carbon. The C/O ratio tells us that most of our carbon comes via the winds from intermediate mass AGB stars. Heavy elements like uranium may be dominantly produced in neutron star collisions, but others like barium and strontium are not.
The details of how many ancestors have contributed to the Sun has no simple answer. Much of the solar hydrogen and helium could be pristine; some will have been through more than one star. Heavier elements (bar some lithium) will have been through at least one star. The fact that we have s-process elements like Ba, Sr, La and Ce, which are formed by neutron capture onto iron-peak elements, tells us those have been through at least two stars.
However, these are vast underestimates. Mixing in the interstellar medium is reasonably effective. The material spewed out from supernovae and stellar winds 5-12 billion years ago has had plenty of time to mix throughout the Galaxy before the Sun's birth. Turbulence and shear instabilities, driven by the winds and supernovae from massive stars, should distribute material on galactic-length scales in a billion years or less (Roy & Kunth 1995; de Avillez & Mac Low 2003), though local inhomogeneities associated with nearby recent events can persist over $10^{8}$ years. If this is the case, then the Sun is the product of the $\sim$ billion stars that died before it was born.
The reason you are confused with your lifetime argument is that you have ignored the possibility of the Sun being made from stars that lived at the same time in different parts of the Galaxy. The material that they ejected near the end of their lives has just been thoroughly mixed up.
$^1$ The rest are produced by the s-process in intermediate mass AGB stars; through nova events on white dwarfs; or perhaps, in the case of the heavier elements, through the collision of neutron stars (see this Physics SE question).
$^2$ A rough expression for the lifetime of a star is $10 (M/M_{\odot})^{-5/2}$ billion years.
