There could be many possibilities. It might be an original character created specifically for the fanfiction. However, it could also be a well - known house - elf from the original series, like Dobby, who in the fanfiction somehow comes to be owned by Harry Potter in a different set of circumstances. Maybe Harry saves Dobby in a more complex way in this fanfiction and as a result, Dobby becomes his house - elf.
Well, perhaps the house - elf is constantly tidying up in really cool ways. It could be folding clothes by magic so that they are always perfectly creased, or it might arrange the books in Harry's library in a way that makes it easier for him to find the exact spell book he needs at any given time. And it could also have a special connection with the other magical creatures around the house, like communicating with the garden gnomes and getting them to stay in line without causing trouble.
The fanfiction could focus on Lemon's backstory. Maybe she was once a part of a different wizarding family before coming to Hogwarts. And in this new environment, she has to adjust to the different dynamics among the other house - elves and the wizards and witches in the castle. She could be facing some discrimination from the other house - elves for being too friendly with the students.
One interesting 'Harry Potter House Elf' fanfiction could be one where the house elf has an adventure outside of the wizarding household. For example, it might discover a hidden magical garden that has been long forgotten. However, I'm not sure about the 'lemon' aspect as it might refer to something more adult -oriented, so I'll focus on the general house elf story. House elves are often underappreciated in the original series, and a fanfiction could explore their individual personalities and dreams more in -depth.
Well, perhaps Harry could start seeing the world from the house - elves' perspective. He might use his magic in new ways to improve their living conditions. For example, he could secretly create better living quarters for them in Hogwarts. He might also try to teach them some basic self - defense magic so they can protect themselves from the meaner wizards.
His personality would be significantly different. Since house - elves have a very different culture from wizards, Harry would grow up with different values. He could be more empathetic towards the underprivileged in the wizarding world, like the house - elves themselves. He might also be less concerned with the pure - blood/muggle - born divide, as house - elves don't really care about such things. Additionally, he would have a strong sense of duty towards those he considers family, which in this case would be the house - elf that raised him. This sense of family and loyalty would be a core part of his personality.
One interesting plotline could be Dobby using his unique magic abilities to help Harry in a new and unexpected way, like finding a hidden magical artifact that could turn the tide in the fight against Voldemort. Another could be Dobby getting into trouble while trying to protect Harry at Hogwarts, causing a series of comical and heart - warming events.
If Harry becomes a house - elf, his character would develop in a way that he becomes more resourceful within the constraints of a house - elf's life. He would have to use his wits to get things done because house - elves have limited magic compared to wizards. Also, his sense of justice would be even stronger. He would be constantly looking for ways to break the chains of the traditional house - elf servitude, maybe by finding loopholes in the ancient laws that bind them.
One plot idea could be that the house - elf imparts unique magical knowledge to Harry that he doesn't get at Hogwarts. For example, teaching him ancient spells passed down in the house - elf community. Another could be that the house - elf protects Harry from threats in a very different way than Dumbledore or his friends. Maybe the house - elf uses its own brand of magic, which is more instinctive and less rule - bound.
Dobby's character could be developed further by showing his growth in confidence. After being freed, he could become more assertive in his actions and decisions, not just relying on his loyalty to Harry but also on his own sense of right and wrong.