It's popular because it gives a new and interesting dynamic to the character of Stiles. Instead of just being the sidekick, he becomes a central figure of care and support in the pack.
It could also be that they are used to Stiles being the one who takes care of them and remembers things for them. He's always been the planner and the one to keep things in order. So they unconsciously assume that he doesn't really care about his own birthday as much as he does about helping them, and they forget all about it.
One great story could be 'The Pack Mom's Guidance'. In this fanfiction, Stiles takes on the role of pack mom in a really heartwarming way. He looks after the pack members, making sure they are safe and happy. Another good one might be 'Stiles: The True Pack Mom' where he uses his intelligence to solve pack problems like a mother would for her family.
You can try fanfiction websites like Archive of Our Own (AO3). Just go to the site and use the search bar to type in 'Pack Mom Stiles baby Liam'. There might be some great stories there.
In many 'female stiles pack mom fanfiction', the theme of sacrifice is present. She may have to sacrifice her own desires or time for the well - being of the pack. Also, the theme of acceptance is there. The pack has to accept her in this new role and she has to accept the responsibilities that come with being the pack mom. There are often storylines where she has to earn the respect of some of the more skeptical pack members, which makes for an interesting read.
One common theme is protection. Stiles often takes on the role of protecting Baby Liam, just like a mother figure would. It shows their bond in a very endearing way.
One great 'female stiles pack mom fanfiction' is 'Stiles' New Role' where she steps into the pack mom role in a very unique way. It shows her dealing with the pack's problems and her relationship with each member in an interesting light.
He might be really hurt. Stiles is a sensitive guy deep down, and having his friends, who are like family to him, forget his birthday would sting. He could go into a bit of a sulk and isolate himself for a while.
The pack mom's relationship with Baby Jackson could be one of nurturing and protection. She could sing lullabies to him that have a hidden supernatural meaning, which only Jackson can sense. As Jackson grows, he starts to see the pack mom as his main source of comfort. And this relationship could also influence how the pack as a whole views Jackson. If the pack mom shows strong affection for him, it could make the other pack members more accepting of the baby, who might be different from them in many ways.
A possible plot could involve the pack mom trying to raise Baby Jackson while Stiles acts as an unexpected mentor. Stiles, with his quick thinking and knowledge of the supernatural, helps the baby develop in a way that surprises everyone in the pack. For example, he might teach Jackson how to control his emerging powers at a very young age. This could lead to a series of adventures where they all have to face different challenges together, like fending off other supernatural beings who are jealous of Jackson's potential.