In the anime, not much is really said about Galaxia's Star Seed, except that she sent her "light of hope" away to a far off land away from Chaos. You see, after Galaxia sealed Chaos inside of her, she already knew it was too late, sending her star seed away was a last effort to make things right. Galaxia knew Chaos was taking over both her mind and body and that soon she wouldn't be able to do anything to stop it. She had made a grave mistake by trying to seal Chaos away by herself without anyone's help.
The Star Seed took on the form of Chibi Chibi and made it's way to Usagi. She was the last remnants of hope Galaxia had in her before her mind was taken over by Chaos. Princess Kakyuu knew of the "Legendary Soldier" who went against Chaos, but not the whole story -- only that the soldier disappeared and left her Light of Hope out there in the galaxy. Kakyuu was searching for the Light of Hope when she fled from Kinmoku and headed towards Earth.
Sailor Moon didn't know Chibi Chibi was the Light of Hope until the end and neither did Galaxia. Chibi Chibi turns into a sword for Sailor Moon to use to fight, but Usagi doesn't want to destroy Galaxia -- she tries her best to avoid battle all together. A completely possessed Galaxia ends up shattering the sword, which in a sense "kills" Chibi Chibi. However, after it is all said and done, the little dark pink heared girl reappears right before heading off into the galaxy after Galaxia.
In the manga there is no mention of a "light of hope" and Chibi Chibi is in no way related to Galaxia.