Unlike her crude daughter Mertle, Mrs. Edmonds is a nice woman to all people, including Lilo and Stitch. It is unknown if she is actually aware of the presence of aliens on the island or simply chooses to believe them as curious creatures who wander around. If she does know they are aliens, she is a lot more open to them than her daughter is and likely enjoys the excitement they bring.
This can be shown in the episode "Clip", where the hair-eating experiment Clip eats the woman's hair after coming into Mertle's bedroom to investigate the noises of Gantu trying to break in and grab the hairball. While she is scared of Gantu and Clip, fainting at one point, she seems to recover from it rather quickly. When Lilo comes to apologize for the chaos she caused, Mrs. Edmonds is not angry at all that her hair is gone and actually believes that the bald style is the "real me." She is even excited when Jumba's hair grows back spontaneously, calling it a spontaneous "something or other."
It seems that Mrs. Edmonds cares for her only daughter deeply and has the best of intentions for Mertle, even if that means spoiling her excessively. She also seems to want her daughter to have kinder friends, which is why she appears to encourage a friendship with Lilo, instead of the mean girls that support her daughter's poor and bossy behavior.
This is seen in the episode "Babyfier", where she offers for Lilo to come and watch the Sludge Mummies movie with them since they have an extra seat. However, Mertle quickly shuts down the idea saying that the seat is for their purses, and Mrs. Edmonds doesn't push it any further. This likely shows that the mother is easily manipulated by her daughter and acts as a permissive parent. This permissive attitude could be to placate her daughter as Mertle often becomes aggressive and throws terrible tantrums when she does not get what she wants.
While we do not see Mrs. Edmonds often, this could be because she has a job being a single mother and may not always have time to spend with her daughter. This could be a factor contributing to Mertle's poor behavior as she acts out to get her mother's attention. This absenteeism or ability to be easily manipulated by her daughter is why she doesn't apprehend Mertle for her terrible behavior more frequently. This lack of attention to her daughter could also be why Mertle does not seem to get the help she needs as she is possibly dyslexic or struggles to read.
While it was not confirmed in the show or movies, it is likely that Mertle's mother and father are separated or divorced. This is hinted at in Leroy & Stitch where when Mertle cruelly states to Lilo that Stitch "ran away and is never coming back," to which Yuki says, "Just like Mertle's dad," to which she snaps angrily for her to be quiet. This separation could be one of the reasons why Mertle is a bully and behaves poorly. Her daughter's attitude and bullying behavior could possibly and partially be attributed to her reaction to her parents' separation and her inability to see her father all the time. Her spoiled personality could also be from her mother spoiling her daughter to keep her happy and mask the struggles they endure as a family.
This can also explain her absenteeism in the show and movies as she is a single mother trying to make ends meet and does not always have time for her daughter.