A star consumes quite a lot of hydrogen in its life, and is pretty much "vacuuming" everything in its vicinity. After it dies (eventually by supernova which will spread all its composition over light years), is there enough hydrogen left in that area to light up a new star? And will that star be more short-lived compare to its predecessor?
There are several misconceptions in your question.
First, a star does not vacuum everything in its vicinity. Rather it forms from a condensation in a gas cloud, which in turn collapses to a proto-star surrounded by a gas disc, which can contribute further material. Once formed in this way, a star typically does not acquire more gas (exceptions are symbiotic binary stars etc).
Second, a star with mass in excess of $\sim8$M$_\odot$ will (typically after a long time) suffer from a supernova, when most of its envelope is flung back into space. That gas is still mostly hydrogen, though enriched by 'metals' (non-primordial elements). However, the gas is hot and fast moving and hence not in a condition to form another star.
Third, the gas from the supernova will eventually mix with other gas and dissolve into the general pool of interstellar medium (ISM). Some of that may cool to form a molecular cloud (as gas cloud where molecular $H_2$ dominates), which in turn may become the site of new star formation.
We know that the Sun has been formed from enriched material, which is a mixture of primordial gas with the ejections of several supernovae.
Our sun is a 3rd or 4th generation star, so yes, there is enough hydrogen left over to create more stars.
We know this because our solar system is fairly rich in heavy elements, which means that there must have been at least 1, and probably 2 or 3 supernovas that created these heavier elements that created all of the rocky planets, asteroids, comets, etc.
It is doubtful that our sun will shed enough hydrogen to create yet another star. It's too small now.
Also, if you look at the pillars of creation, which is a nebula created by a supernova, you can see the early stages of star formation happening right now.
First, thanks to @LCD3 for leading me on the right path here. My original answer was inaccurate, and so I got rid of it.
A supernova occurs when a very massive star can no longer sustain enough nuclear fusion to combat the force of its own gravity pushing inwards on it. This happens after the star has gone through different stages of fusion. Typically, it begins with fusing hydrogen into helium. This is the type of fusion you have probably heard about the most because stars are largely hydrogen and helium. However, there are other fusion processes that are equally important when it comes to prolonging a star's life, which fuse together heavier elements.
A star begins by fusing hydrogen nuclei into helium nuclei deep in its core. This is how the star produces energy, and is indirectly responsible for the star shining. However, there is only so much of this fusion a star can undergo in its core. When the core hydrogen is depleted, the star beings fusing helium there. It continues hydrogen fusion in its outer layers, where there is still hydrogen. Eventually, the star runs out of helium in its core, and begins to fuse even heavier elements. Hydrogen fusion continues in the outermost layers, with helium fusion occurring in lower layers.
Unfortunately, the process can only go on for so long, and eventually the star can't fight gravity any longer. In very massive stars, this leads to a supernova, which flings off much of a star's mass into space. In all the cast-away matter, is there enough hydrogen left to form a new star? Well, there isn't nearly as much hydrogen as there was at the star's birth. In relatively low-mass supernova progenitors, there might not be enough hydrogen to form a new star. In very high-mass stars, however, there will still be a substantial amount left. Could this form a new star? Probably not for a long time, because the hydrogen will have been thrown off into space by the supernova, and the wouldn't be very dense. It wouldn't be easy for it to collapse into a gas cloud to form a protostar. I wouldn't rule this out for very high-mass stars, but in the remnants of many stars, there probably would not be enough hydrogen to form a new star.
I hope this helps.
Source for the layer explanation: http://www.astronomynotes.com/evolutn/s5.htm. Also, many thanks to @LCD3.