1. Initial Motivation:
First, establish the androids' primary directives:
- The preservation and support of the human crew.
- The preservation and support of the spacecraft.
- The preservation of the ship's exploration data until retrieval by authorized personnel.
Directive 1 has failed. Directive 2 is impossible given the state of the ship. Directive 3 is all that remains.
Because this directive persists until more humans arrive, it incurs an implicit sense of self preservation.
The ship was designed for very long-distance/long-term trips, and would reasonably have a way to fabricate replacement components. This fabricator would probably not be able to handle any raw material (eg, you can't throw in a lump of iron ore and expect it to work), but would probably be able to use materials of a certain purity which would've been stored for this purpose--similar to how 3D printers need spools of a specific type of plastic.
This would allow the androids to produce replacement components for themselves to prolong their own lives. They could also produce the necessary components to create backup fabricators. Eventually, though, they would run out of the necessary materials for the fabricator.
Upon reaching a critical lower threshold of fabrication materials, the androids would begin to create ways of refining raw materials into substances that the fabricators could use. They could derive this information from reference materials on the ship's computers, and their own understanding of the specs of every device on the ship.
Eventually they would have machinery capable of converting any available substance into useable fabricator materials. There would still be components which would require rare elements they couldn't readily obtain, but they'd essentially be able to skip much of the "stone age" technological requirements.
The next step is the jump from "preservation of self" to "preservation of species". The androids would realize that there are still situations which could incur an irrecoverable loss of one of their number, and therefore jeopardize the entire mission.
The solution is the same as it was for the fabricator: redundancy. Instead of merely creating replacement parts for themselves, they would work to create duplicates of each member, and of the exploration data itself.
They would eventually need to cannibalize the ship's own computers in order to grow their numbers past a certain point (producing the processors necessary for their brains purely from raw materials is still beyond their capabilities); from here on, each android will contain a copy of all exploration data within themselves.
Internalizing the data which is the soul of their entire purpose has an unintended consequence: it makes each android life important, to a degree. Instead of duplicate androids being somewhat expendable, the loss of a single one represents the loss of an entire copy of the data they are collectively meant to preserve. If data is corrupted over eons, then comparing the copies is the only way to ensure its integrity.
Perfect duplication carries with it several problems: blind-spots and deadlocks. Essentially, if the original androids were unable to solve a problem (such as synthesizing the necessary materials to mass-produce their brain processors), then no number of duplicates would be able to solve that problem.
Similarly, it was possible for two duplicates to find themselves in logical deadlocks between each other until external stimuli broke the loop.
The solution was to add a degree of randomness to each new duplicate. The original directive could not be changed, and neither could the exploration data, but most of their personality and cognitive functions could be tweaked slightly.
Whichever duplicates showed the most beneficial improvements would be used as a baseline for future generations, as decided by the un-altered Originals (with the intent to ensure that the alterations didn't drift in any direction which could harm the mission).
At this point, their population was still limited, so the minds of duplicates were successively altered with each "generation", as they worked towards the goal of cracking mass production. Bit by bit, their ability to conceive of new solutions improved. Their technological capabilities grew step by step.
After many centuries of tiny improvements and gradual degradation of their population (the existing brain processors could only be repaired so many times before total failure), one of the highly "evolved" duplicates managed to determine a method of synthesizing the necessary raw materials for their processors.
From that moment on, their population was no longer limited, and they spread across the globe. Their species grew and continued to evolve, but within each of them is still a copy of the original exploration data, ostensibly unchanged after billions of duplications.