I’m eager to see why the Sun Incinerator means so much to Emerald. Here’s my theory of the reason:

Ships and their Gem owners are shades of a colour shared between them; we’ve seen this with Aquamarine and her ship and Emerald and her Destiny Destroyer. The Sun Incinerator is a Gem ship that is shades of yellow, brown and dark green. A Heliodor, a beryl like Emerald that could be a shade of yellow, brown or yellow-green, could’ve been the original owner of the Sun Incinerator. Heliodor means “gift of the sun” in Greek and the Sun Incinerator’s hull is shaped like a hexagonal prism, a possible reference to Heliodor [and the rest of the beryls] belonging to the hexagonal crystal system.

Potentially, Emerald and Heliodor were close friends and the former gained ownership of the Sun Incinerator as a keepsake after the latter was shattered in The Rebellion. That would explain why Emerald had the Sun Incinerator guarded by 67 elite Citrines and looked mournful when she told Lars that stealing it was “unacceptable”.