Future Press game guide description: "This grouchy old woman lives near the Dry Dock in Central Yharnam. Like most other Yharnamites, she distrusts outsiders and will make a show of belligerence when you first meet her, but will nevertheless ask you if you know of any safe places where she can take refuge during the coming night's hunt."
The Lonely Old Woman is a Yharnamite and a lot like the Bigoted Old Man in disposition and opinion. She is distrustful of outsiders to Yharnam, blaming them for everything wrong that has happened there. Despite this obvious belligerence towards outsiders, she will accept the player's advice about a safe haven, likely out of necessity, since she's heard that: " shutting up indoors isn't always enough." and because there is a deformed dog barking away outside her house, likely trying to get in when the player first meets her.
If the player tells her to ascend to Oedon Chapel she will next be found sitting on a chair there. The Chapel Samaritan will remark that she doesn't offer him much in the way of conversation, indicating that she has the same biases towards him as the Bigoted Old Man and Adella. The Lonely Old Woman is entirely ungrateful for the player's help and will excuse herself from any recompense by blaming the player for being an outsider who caused the Yharnamite's blood to be ruined, "tainted by your ilk!"
The Lonely Old Woman's dialogue indicates that Yharnamites do not understand the source of the Scourge of the Beast. Instead of blaming the Healing Church, which they may have done to begin with, it seems that the Healing Church passed the blame on to outsiders to Yharnam, a simple diversion that deflects blame onto a group of people who cannot defend themselves from the accusations, since not all foreigners to Yharnam came together or from the same place.
If spoken to later on in the evening the Lonely Old Woman will have entered a state of depression. She is reflecting on how Yharnam used to be, before the Hunt and the Scourge of the Beast overcame it. She will remark "Oh, the good old days, what a laugh, eh?" reflecting on when blood was first introduced to the populace, and the wonders that it achieved, perhaps these thoughts guided her towards a revelation…
Liquid medicine concoted at Byrgenwerth.
Calms the nerves.
Those who delve into the arcane fall all-too-easily to
madness, and thick human blood serves to calm the frayed
nerves of these inquisitive minds. Naturally, this often leads
to a reliance on blood ministration.
As time passes the Lonely Old Woman will begin self medicating with Sedatives. Sedatives are, as the description states, thick human blood, a description very reminiscent of Thick Coldblood "which is doubtless, the product of obsession, a potent source of human strength." It seems that the Lonely Old Woman had a significant amount of sedatives that she brought along with her. While the majority of Yharnam are heavily reliant on blood, the use of sedatives is most closely associated with the scholars of Byrgenwerth.
Scholars of Byrgenwerth began taking thick human blood as a sedative because their inquisitive minds were delving into the arcane. The arcane is heavily associated with induced frenzy from investigation into the Great Ones in the Old Labyrinth (or Chalice Dungeons) below Yharnam. From "The Paleblood Hunt" by Reddit user Redgrave (https://docs.google.com/document/d/1JL5acskAT_2t062HILImBkV8eXAwaqOj611mSjK-vZ8/preview):
[…] “Frenzy is a condition which occurs when an individual is forcibly made to wrap their minds around the eldritch nature of the Cosmos and the Great Ones. To put it another way: Insight is what happens when an individual slowly learns, discovers, and processes the information of the Great Ones. Frenzy is what happens when an individual is forced to process the information.”
We do not know what caused the Lonely Old Woman to self medicate, but perhaps when reflecting on her past she began to realize that the source of Yharnam's problems was not the outsiders, but the blood. This kind of information would certainly drive one towards despair, and to cope she ironically uses blood.
After she begins medicating the Lonely Old Woman will mistake the player for her child. From her dialogue we can gather that her child was frequently “in a bind” and that she would provide them with sedatives to ease their suffering. This naturally led to a reliance on blood ministration and we can see the effects of this the more we talk to her. Her dialogue becomes more frantic as she remembers her child growing more and more violent and impatient, obviously undergoing the effects of the Scourge of the Beasts.
From this we can gather the sad story of the Lonely Old Woman. Both her child's father and her child have long since died, leaving her alone and somewhat helpless. Her child likely transformed into a beast, but what about the father? The only reference we get to him is when the player kills the Lonely Old Woman. She will say: “…It’s your father’s blood…” What she means by this is of course never explained. One possible interpretation is that she blames the streak of violence in her child on their father's blood. Perhaps their father was an outsider to Yharnam, perhaps even a hunter. This may be the root of her vehemence towards foreigners to Yharnam.
Another possible interpretation is that the blood she has been providing to the player in the form of sedatives is the thick coldblood of the child's father, all in bottled form. This may mean that the father was a scholar at Byrgenwerth before passing on, but that is speculative.
If the player shares their woes with the Lonely Old Woman she will go and find them some sedatives. Perhaps at first she provided them with her own personal stash, giving them more and more as a reflection of the demands her child made on her as their thirst for blood grew. Eventually she will run out and will have to go out searching. Sedatives are certainly not easy to come by in Yharnam, it would make sense that this woman is returning to somewhere she knows she can find sedatives; her home. If repeatedly asked for sedatives, the Lonely Old Woman will eventually be found dead before a gravestone nearby Oedon Chapel. The likely conclusion is that she was killed by some ranging beast on her way to find more sedatives, but perhaps the gravestone she lies before has more significance to her than we realize. Perhaps this is the gravestone of her child, and the realization that they are dead pushed her over the edge. Whatever the case, the Lonely Old Woman meets a sad fate not wished upon anyone, even one as insulting as her.