I think Esther is a lot more of a complex character than people want her to be. It's easier if she can just be painted as an awful mother. But it is clear Esther did love her kids, all of them. Just differently. I don't think you can truly measure love in quantity, but she definitely had favorites and least favorites.
Esther loved Freya so much that losing Freya fundamentally changed who she was. She wanted to be a mother so badly that she was willing to do anything to make it happen. People often blame her more for trading Freya for the ability to have children, but don't blame Dahlia for making her own sister trade away her first child. Esther was young when she made that deal and didn't fully know what it meant. She also probably naively believed her sister wouldn't steal her child. But she loved Freya dearly.
Esther also loved Finn a great deal however their relationship grew twisted and toxic because of the loss they both shared. She essentially trauma bonded with her four year old child. Finn's trauma is rarely discussed but it is clear that his mother became his safety after losing Freya. He would do anything for her and Esther exploited this at times and even used him for emotional support. She did love Finn, but the trauma colored that love.
Elijah is her least favorite child. She basically admits it when she tells Klaus that he brought joy to their lives and made them a family again after losing Freya. She was heavily pregnant with Elijah when she lost Freya meaning Elijah's birth and multiple years of life did not bring their family joy or make them feel like a family. Although it's important to remember that while pregnant, she lost her child and moved across the ocean to a new land where she was alone with a volatile husband as this was also the period where Mikael starts becoming a bad/potentially abusive husband. I can't imagine she didn't experience some level of post-partum depression after having him. To top it off, Elijah favors Dahlia more than either of his parents. He likely reminded her of her sister and everything she had lost.
She views Elijah as more of a second parent. She expects him to always behave and keep his siblings in line. When the others step out of line, she has more sympathy for them. She offers Finn, Kol, Klaus, and Rebekah second chances in witch bodies but tortures Elijah, mocking him for what he is and that he is no longer her "noble son." We never see any exchanges between Elijah and Esther that have any type of warmth. This doesn't mean she didn't love him, but hey have a very strained/neglectful relationship.
Klaus is her favorite child. He was the only child she had with the man she truly loved. Even though Klaus seemed to get the brunt of Mikael's anger, he also gets the most protection from Esther. To the point that Finn is jealous of this attention. Klaus arguably did the absolute most vile deeds that should disgust Esther but she still showed love and compassion to him. He was the problematic child that could do no wrong.
Kol is pretty low on the ranks but still above Elijah simply because he had magic. Esther would have wanted to pass on her magic somewhere even is she wasn't truly practicing. She would have loved it but also feared it. Kol would have also driven her crazy because he was always so independent and often getting in trouble.
Rebekah has a special place in her mother's heart as the only daughter she had after losing Freya. It would have been painful at first as she looked so much like the child Esther lost, but she would have viewed it as a second chance. Rebekah always had a soft spot for Esther and losing her mother devastated her.
Henrik is pretty high at the top simply because he is the youngest. Henrik seemed to have a happy spirit from the few brief moments we see him. The baby of the family is always special. His death sent Esther into such a spiral that she created vampirism.
None of this means I think Esther is an awful mother. There are definitely terrible things she does, but at the end of it, she truly loved her children. Was she the best mother in the word, no. But we have to remember she was a young woman who was also a victim.
