Abyssal safe spots are tactical bookmarks created by using Abyssal filaments. Capsuleers utilize deep Abyssal safe spots to hide from other players.
Be disconnected by downtime while inside of any Abyssal area.
Normally, disconnecting while inside of an Abyssal area results in ship loss due to incoming damage or flying out of bounds. However, if one's ship survives a regular disconnect, the area can still be finished after reconnecting. But being disconnected during downtime instead transports one's ship back to the solar system from which they entered the Abyssal area.
The location of one's ship after logging back in is semi-random, but it follows a pattern.
All Abyssal safes sample from a disc shape based on the system's size, or the distance from the sun to the farthest celestial. The disc can extend a significant distance past celestials on the X/Y axes, but the disc's Z axis is always less than the X/Y boundaries and isn't related to celestial positioning. This is demonstrable in systems with celestials with odd Z positions.
This Abyssal disc shape is just the boundary, one's ship can spawn anywhere in the disc, even right next to celestials or in otherwise-undesirable locations. This means there is a chance to "fail" creating Abyssal safes.
This is a solar system saturated with Abyssal safes. The first image is from a top-down perspective, the second is from the side. Note the disc shape.

Further study is required on the algorithm responsible for the disc's shape.