This is referenced in the Star Wars: Rogue One - The Ultimate Visual Guide. In short, Lyra Erso has developed a spiritual connection to Jedi philosophy and may be a covert Jedi sympathiser.
Although Lyra has never met a Jedi Knight. she studied their history and philosophies, even when the Galactic Empire frowned upon such research. It is not known if she ever made contact with the rumoured Church of the Force movement reputed to be growing in the shadows of Coruscant, but she did at that time don the scarlet vestments of the sects native to Jedha - One of the last holy outposts of Force knowledge in the Galaxy.
The crystal necklace was indeed a khyber crystal. It has some Aurabesh writing on it, but it doesn't appear to have any meaning when translated into English (UT?N THN)
This small stone - a fragment of khyber crystal from Galen Erso's research - caught Lyra's eye as her family fled Coruscant. She took it to be a good omen and passed it to Jyn, who at the time didn't understand, but knew it was a gift nonetheless.