The resting focal length of the eye is around $17$mm - you see various different figures but let's assume $17$mm is correct as the exact value doesn't affect what follows. Anyhow, that means the distance from the optical centre of the lens/cornea to the retina is $17$mm so in the lens formula we require $v = 17$mm for a clear image.
The eye focuses by deforming the lens to change the focal length. If we take the object distance to be 250mm i.e. the least distance of distinct vision and use the lens equation:
$$ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} $$
We find that for a clear image at $u = 250$mm and $v = 17$mm the focal length of the lens has to be reduced to about $15.9$mm.
As we age the lens gets less flexible so our ability to reduce its focal length decreases. If we use your value of $500$mm for the least distance of distinct vision of an elderly person, i.e. $u = 250$mm and $v = 17$mm, the focal length of the lens works out to be $16.44$mm.
So if you want to allow the position of closest focus to be moved in to $250$mm you need spectacles that increase the ability to focus. Since there is quite a large distance between the lens/cornea and the spectacles you'll need to do the calculation for two separate lenses, but this is straightforward. Calculate the image position from the first lens then use that as the object position for the second lens.
The term "focusing" means something else than the OP suggests. It means that the different light rays coming that a single, specific point $P$ of the object emits to different directions re-converge back and reach the same place of the retina.
If the object, and/or its point $P$, is infinitely (very) far, then the light rays coming from $P$ are (nearly) parallel near the eye. But even if they're divergent, not parallel, the eye is able to refocus them so that they reach the same pixel of the retina.
So the single convex lens don't magnify anything. At most, they do exactly what the lens in the eyes do and what is needed for focusing – to redirect the nearly parallel light rays so that they intersect against less than an inch from the eye's surface, on the retina, again.
The diagram posted above is misleading because it suggests that the two parallel rays that are supposed to converge to the same point of the retina come from different ends of the objects we observe, $P_1$ and $P_2$. But that's not the case at all. If the diagram is fixed so that it makes sense, we see just one point $P$ and both (nearly parallel) light rays originate from the same $P$.
If we want to consider two points $P_1$, $P_2$ of the object, as the bottom part of the picture clearly wants, they must create two distinct dots $Q_1,Q_2$ on the retina, i.e. two different intersections of pairs of light rays!
So again, a single convex lens doesn't magnify anything. It just does what the eye has to do to focus, anyway: to make the light rays converge. To calculate whether an arrangement of lens (and yes, at least two lens or lens+mirrors are needed) are able to magnify, one has to consider the size of the image on the retina, i.e. different intersections of the light rays on the retina, separated from each other. To approximate the eye by a single dot isn't good enough to calculate the magnification!
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Great question! I suspect it's a research problem.
My brother-in-law, Chris Croke, and R. A. Hicks, Designing coupled free-form surfaces have shown how to do particular image transformations with mirrors, and the paper says it is also possible with lenses.
Quoting from the abstract:
Here is a pair of mirrors that rotates an image by 45 degrees:
And here is the result from a ray-tracer: