Assuming that it can't, how is it to compete with the other neurons? (There's a school of thought that neurons compete with other neurons for resources by making useful connections.)
But replicating the original connections isn't necessary, and certainly not necessary for the neuron to "compete." For example, say an old corporation dies, e.g. Microsoft . Could a new corporation replicate the 1000s of precise products, services, and contracts of the original? Not at all. But replacement corporation could find new things to offer, potentially taking advantage of the economic shortfalls and new demands precipitated by the demise of Microsoft. Eventually the economy returns to full efficiency, though in a slightly different form than before. And just so with neurons- the information passed through the neuron that died will now be in demand by neurons downstream, and this demand can be fulfilled by new connections by the new neuron (or new connections from old neurons, or simply not.) But the information may come from different places, or have slightly different semantic value, or other differences.  Brains, like economies, change, and are good at changing gracefully.
 At least, as far as we know- with glial cell participation and functional viral capsids and whatever else is going on, anything is possible, I guess.
 Example chosen without malice. Honest!
 All assuming the competition model of neural organization. Which is a plausible model last I checked, but see .
We know that there is lots of redundant networks and plasticity, so if one cell goes down does it matter whether the gap gets filled by and old vs. new cell?
With 100 billion neurons we seem fine with what we have. The bigger question seems to be how to protect these 100 billion cells from degenerating.
Yes, on a lot of fronts, but strokes, in particular, would be a lot easier to fix if the brain could renew itself. Fixing brain problems becomes much harder if the brain never regenerates.
> With 100 billion neurons we seem fine with what we have. The bigger question seems to be how to protect these 100 billion cells from degenerating.
Maybe. But we also know that those who age well seem to recruit cells from a different layer of the brain as they age. This would seem to indicate that the brain is having to "make do" for some reason which we do not understand.
In addition, what is "part of the brain"? Are the optic nerve or your olfactory nerve part of the brain? What about your spinal column? The computer scientist in me says "Of course" given how strongly linked they are with the processing.
Having a known ability to rebuild neural pathways would be important.
"The Brain That Changes Itself: Stories of Personal Triumph from the Frontiers of Brain Science"
Prevention, as you mention, is a market.
Repair, is also a market.
Of course there are other factors.
That is a relative question. I imagine it really does matter to a variety of researchers in various fields. At the very least, it matters to those involved in so-called basic research.
Might it be possible that some brains have the ability to generate new neurons, while others do not? Couldn't it be possible that evolution created such situations?
Sure. For instance, female ova. The stock is limited.
> Might it be possible that some brains have the ability to generate new neurons, while others do not?
The former is obviously the case for rodents at least. Could it differ within the species? I can't see why not, in principle.
> Couldn't it be possible that evolution created such situations?
I imagine it's more about gradual loss of function. For example, in larger brains it's progressively harder (and less probable) to transport the neuroblast to the area where its growth would be beneficial for the brain at large. Thus, eventually adult neurogenesis ceased to increase fitness in large mammals, became effectively an atavism, and mutation load disabled some relevant genes, but it still could hang on a few pathways that either work or do net depending on specific alleles present in the population.