The image will be recompressed. The two scenarios you describe are actually effectively the same, because the lossy part of the JPEG compression discards information which stays gone when the image is decompressed. (Hence, lossy.) That means that reapplying with the exact same parameters shouldn't do much, either in terms of further space saves or in terms of further artifacts. The differences come down to precision and rounding errors. (This is the same in Lightroom as it is in any other program.)
So, if you recompress with exact same parameters and have aligned your crop to 8×8 blocks, the degradation should be minimal. However, if you're using a high level of compression (I think 80% qualifies), you might actually see a difference, because the artifacts introduced by the initial compression are permanent changes to the image and will get recompressed too, possibly causing more artifacts.
Setting to 100 will be more safe, as any newly added artifacts will be hard to notice. It won't make the image any better, but not significantly worse. However, it will introduce changes across the whole image, whereas resaving will mostly concentrate changes to where artifacts are already noticeable. This, unfortunately, means that your mileage will vary.
If you're resizing or have made significant manipuations, all bets are off.
As a quick test, I used ImageMagick to recompress a JPEG image over and over at 75%. The samples below are uploaded as PNG files to avoid yet further recompression, and were doubled in size when I converted to PNG to make the effect more obvious. It turns out that after eight resamplings, the effect converged on a perfectly stable result, where recompressing again results in a bit-for-bit identical file.
Here's the uncompressed original:
Here's the result of going to 75% JPEG:
And here's that resaved:
Woah; that's actually much more noticeable than I expected, and I work with JPEG all the time.
So here's the final converged image (8th pass):
Again, colors are definitely even more off, including some false color patterns, and the blocky artifacts jump out more. So, don't do that.
But here's the same thing with a 99% quality level, after 9 passes (the point where it converges so further passes are identical):
Here, the difference barely registers. (I mean that literally; compare them pixel by pixel to the non-compressed version and the deviation is just very slight random noise.) So, what if I go back to that first 75% image and then resave at 99%? Well, this, (after just once):
That's definitely visibly better than resaving as 75% again, kind of to my surprise. But, there's obvious new degradation around the pink trimming and the eyes. With the recycled version of the same settings, the JPEG artifacts are being exaggerated with each recompression. With the low resolution and low quality I've chosen, that turns out to be worse than recompressing everything differently.
But, it's also worth mentioning that saving 75% one time is much worse than resaving at 99% a million times. In my example case, the artifacts at 75% are so obvious that the further degradation is like dumping water in the ocean. If you save at a high enough level that these artifacts are not really visible, then my original advice (same settings, will yield basically the same result) still applies. Of course, if you can stick to always working from uncompressed originals, you're better off.
If you can't, I'd try it both ways and see what looks better for your particular image.