Bilhah was a sad child and it was easier to leave her alone. She rarely smiled and hardly spoke. (1.1.45)
Not only is Bilhah the last child of Laban, figuratively placing her last on the totem pole, but she's also the orphan child. Her mother abandoned her at a young age, and she's the lowest born out of the daughters—her mother was a slave. Bilhah has very little going for her: "Not even my grandmother," says Dinah, "[…] could warm to this strange, lonely bird, who never grew taller than a boy of ten years, and whose skin was the color of dark amber" (1.1.45).
Okay, so Bilhah was a pretty depressing child. Even her own grandmother didn't seem to love her. But as Bilhah grows into a woman, she becomes one of the more pleasant and loving characters in the novel. When she senses Rachel's sorrow, for example, which stems from her miscarriages, Bilhah says:
"Let me bear a son on your knees. Let me be your womb and your breasts. Let me bleed your blood and shed your tears. Let me become your vessel until your time comes, for your time will yet arrive. Let me be your hope, Rachel. I will not disappoint you." (1.3.2)
Despite being ignored as a child and pretty much passed off as a slave, Bilhah shows tremendous strength and love. Though this favor that she would be doing for Rachel is a bit grounded in self-interest (as Bilhah wanted her own child), she still exhibits more sisterly love than any of the other sisters. And that's all the more remarkable given the fact that she wasn't even treated as a sister when she was growing up.
Even to Dinah, Bilhah is the loving auntie who can make her sadness disappear. When Leah scolds Dinah for throwing her spindle on the ground, for example, Dinah goes to Bilhah to feel better. And that's just what happens—Bilhah tells Dinah a story to warm her heart:
I smelled the soft, loamy musk that clung to my youngest auntie and listened to her sweet, liquid voice and forgot all about the ache in my heart. And when her story was over, she showed me that the string on my spindle was as evenly made and strong as Leah's own handiwork. (2.1.31)
Bilhah is the one who often lets the other characters forget about the aches in their hearts. Though no one showed her compassion as a child, Bilhah nevertheless treats everyone with great reverence. She's forgotten and mistreated, yet kind and loving. Maybe being mistreated for so long has helped her to understand and comfort other people's suffering.