There are so many talented writers out here, who I trust with the characters far more than I do myself. Please feel free to take over this head-canon that's been tumbling around in my brain and make it your own:
When Elsie was a housemaid, she found herself in the family way. There are many, many possibilities that this could have occurred. One scenario that I came up with is that one of the house's young gentlemen was very taken with her. I imagine him to have had his heart in the right place, but was completely oblivious regarding the very real power dynamic he held over a young Elsie.
Despite her close relationship with her superior servants and the pleas of the young man, the aristocratic family sent her away some sort of home for unwed mothers. Not only does this factor largely into Mrs. Hughes' views about how to keep a certain distance from her employers, but because she was a young adult, some of things that were said to her shaped her own expectations for younger generations of women; echoes of them come back throughout the 19-teens and 1920's with the girls she supervises.
Though Elsie wasn't working at Downton during this time, the chronology correlates with the early years of Cora and Robert's marriage. Cora is trying her hardest to be accepted by the other old families, and is going along on with a few other women of her set to do charity work at the organization where Elsie has been shushed off too. Cora happens o be pregnant with Mary, and the two women form a special bond.
There's plenty of room for your own ideas as to whether Cora continues to return and what the two women are able to talk about. Ultimately, I imagine that Elsie can't help compare Lady Grantham's circumstance to her own. Perhaps she meets the infant Mary shortly after she's been forced to give up her own child, without ever getting to hold her own baby. Or perhaps even a young Mary holds all those characteristics of the woman forced that fate on her in the first place.
After the profound effect these early experiences had on her, it's no wonder Mrs. Hughes expects to have to take care of herself when she faces the possibility of having cancer. However, Cora was also profoundly affected by her encounter(s) with the younger Elsie—perhaps one of the first women in England to show her any kindness—and would never let face such an unimaginable circumstance without support.
Many blessings all—do wonderful things!
(Note: Please do keep the Beschdel Test in mind!)