Hello! My name is William, from London. Since yesterday, when I first became aware of this case, I've caught the sleuthing bug and, after a few hours, I believe I may have found Jane Doe's family. It's certainly a bit of a long shot.
My theory is that Jane Doe is the daughter of Eric Alexander King and Lorene May Wadleigh. Intuitively, I thought that there was something off-colour about the name 'Lori Erica Kennedy', as if some sort of answer lay there, hidden in plain sight. My first thought was that it signified something from her past, perhaps a play on words. I also did not believe that Jane knowing about Becky Sue Turner was the result of travelling across states and hours spent mining public records; I imagined something more mundane: she knew about Turner because she lived in the same area as the cemetery. Finding a post on here, from 2013 (by neesaki), where an Eric A. King is mentioned, only encouraged this idea. So, after performing searches on ancestry.co.uk, I found these two individuals ("Eric A. K..." and "Lorene"). Here are the facts as I know them:
1) Eric Alexander King was allegedly a stockbroker from Rathdown, near Dublin, Ireland. He was born on 25 September 1936, had an older brother named Malcolm, and married Lorene May Wadleigh (daughter of Donald Jacques Wadleigh and Mildred Lorene Canfield) on 1st July 1961 in Seattle, Washington.
2) Eric was naturalized as an American on 1st May 1985. At the time of his naturalization, he, and presumably his wife, lived at 1530 Vista Drive Southeast, Puyallup (near Tacoma), which Google informs me is 2.9-3 miles away from Woodbine Cemetery, Puyallup, where Becky Sue Turner is buried. Lorene continued to live in the area of Tacoma until the early 90s. She later remarried after Eric's death.
3) Lorene was born 26 October 1937 in Kansas. She lived in Seattle and Auburn (not simultaneously) in the 90s. The name "Lori" can be used as a diminutive form of Lorene. The witnesses to Lorene and Eric's marriage were Malcolm J. King and Judith Richardson.
At this point, I do not know if they had any children. Ancestry.com (and .co.uk) has not made California Births for the 1960s publicly available. I would have to order a birth certificate from the Washington State Department of Health, but I don't know if the protocol for ordering certificates is different in the US to how it is in the UK. I believe it's worth a look, given the similarity of the names, the timing of the marriage in 1961 (if they had children soon or sometime afterwards, it would suit Jane Doe's approximate age) and the close proximity that Eric and presumably Lorene's residence had to Woodbine Cemetery.
Sources (requires ancestry.co.uk or .com registration to view original record images, except for FamilySearch results):
Eric A King in the Ireland, Civil Registration Births Index, 1864-1958
Malcolm J King, Ireland, Civil Registration Indexes (FamilySearch)
Eric A King in the Washington Death Index, 1940-1996
Eric A King in the Web: Tacoma, Washington, Obituary Index, 1939-2010
Lorene M Wadleigh [and Eric A. King] in the Washington, Marriage Records, 1865-2004
Lorene Wadleigh in the 1940 United States Federal Census
Eric Alexander King in the Washington, Naturalization Records, 1904-1991