A Launch Loop is an iron cable just 2 inches thick that runs inside of a sheath at tens of thousands of miles an hour. The sheath has electromagnets inside of it to accelerates the cable. The motion of the cable would cause the loop to curve and raise into the sky.
To outside observers, it would look like a giant version of the St. Louis arch with the peak nearly in low earth orbit, but only a few inches thick. The magic comes with the fact that the iron cable can be used for travel by using electromagnetic attraction to the iron cable through the sheath.
What's interesting is that while there are several engineering challenges, unlike a space elevator, this wouldn't require any materials technologies that don't already exist today. The iron cable could be made by existing forges that make the cabling for bridges, and the electromagnetic sheath could borrow a lot of existing technologies from the hard drive manufacturing industry.
It would also fit with his previous work on Space X and Solar Cities to some extent.
A launch loop can be used to transport cargo or passengers high enough that the energy required to get them into orbit is very inexpensive, which ties into his Space X work.
The external sheath of the Launch Loop could have solar panels on it to power the electromagnets that help keep the loop in motion, and even when the sun goes down, millions of tons of iron spinning like a flywheel has enough kinetic energy to keep going until the sun rises again.
I wouldn't be surprised if he ran across the concept when he was researching ways to get into space before he founded Space X, and it's his "Step 2".
Lastly, if you had a giant cable stretched across the sky that worked kind of like a bicycle brake cable in its sheath, but the cable was running through the system at 30k mph in a giant loop, what better name than Hyperloop? It also explains why a trip that he describes as being twice as fast as an airplane would take 30 minutes. It takes more time to go to LEO and back down
Keep an eye out for Musk buying any patents on electromagnetics or him buying a oceanic cable-laying ship.