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Imagine that we start expanding the International Space Station until it becomes a ring encircling the whole Earth. Then we started widening the ring until it met at both poles, forming an orbital eggshell around the entire planet.
Ignoring all the problems, like that it's starving the planet's surface for sunlight. What would you call this kind of enormous space station?
A Dyson Sphere is meant to capture radiation and convert it to usable energy. Since an earth-like planet only reflects light back, that point would be moot. In any case, here's some suggestions:
Well a Dyson Sphere is named after the person who popularised the idea and also the shape of the object itself.
Without paying homage to a person or entity, it makes sense to call it after its function. Around a planet, there's a few functions it could have which allow you to derive a name for it.
Another way of looking at this, if the orb has an atmosphere of its own, it's plausible that no one is even aware that there is a planet inside the sphere. This would make an excellent set piece, where the name would simply be whatever the Planet was called.
What I'm saying is: There is no widely accepted term for it. Go and be the person to make it!
There is no name for the sphere version. The ring version is an Orbital Ring.
You could most likely use the term "Planetary Dyson Sphere" or if it is defensive in nature "Planetary Shield". I've also heard the term "shell" used to describe structures similar to what you are asking about.
Be aware though that it is a structure with no real value and yet incredibly expensive to build. Orbital rings are very useful but a full sphere not so much.
You could call it a roof. Or perhaps an armilla?
If you were going to get all hard SF about it, a rigid shell around a planet (or indeed around a star) wouldn't be stable, and would indeed fall onto the planet unless it was perfectly centered on the planet's center of mass as @RichardTingle suggests. If it were as low as the ISS, I doubt it could last a week without some kind of (huge) thrusters to keep it in place.
A ring is OK because each part of the ring is in orbit. But as you start to widen the ring, the edges are not in orbit-- they're moving too slowly and in the wrong direction for their altitude, which means they're pulling on the rest of the ring, and it would be nearly impossible to balance those forces.
However, an airtight sphere could be supported by air pressure, and have tethers/struts connecting it to the planet. Those wouldn't have to support the weight, they'd just be to stop it drifting around. Could be a good source of cover artwork...
The specific instance of the construction, built around Earth specifically? I wouldn't be at all surprised if it were referred to as The Firmament.
The general construct I might refer to as a hoover bag, if I were feeling flippant.
It would be called "Weird Transit Feature" ...by aliens! Its purpose is to make Earth both more easily visible and more intriguing for alien civilizations who search for other planets using the transit method.
Earth will become more visible when transiting the Sun, because its larger radius would produce better noticeable transits. Also, the transits would be more interesting because mass/radius/density/gravity calculations performed by the aliens will show that this planet is something very unusual, deserving more detailed investigations.
So it is it. We just need to build this thing and then wait to be visited by our interstellar tourists!
The closest thing I could find is the Globus Cassus. It is a mega-structure around the planet Earth, but it eventually uses up all the matter of the earth.
Supramundane shell or supramundane world is the established term, a concept introduced by Paul Birch, originally in his article "Supramundane Planets", Journal of the British Interplanetary Society, 44, 169 (1991) [PDF].
The term is derived from the Latin supra "above" and English mundane "of, relating to, or characteristic of the world", which was in turn derived from Latin mundus "world".
It's called a Shell World. What you're describing isn't exactly like what's referred to in this article, but I think it close enough to serve your purposes, and it might give you some ideas.
https://www.space.com/23082-shell-worlds-planet-terraforming-technology-infographic.html
