Hestia- Born not of Blood, but by accident. She gathered the energy from a loving Family and used it to birth a child, like Athena using mortal ingenuity... or Perhaps using a Spermbank?
The Child would be beautiful, with Brown Hair and redish brown eyes. She would smell like smoke and good food...
From her mother, she would inherit the Ability to radiate peace and tranqulity, Grant Sanctuary from the Gods or bless a home with such a power, sense the Connections between family, Summon The Hearth fire not to Hurt but to purify and sacrifice, cook incredibley welll and so on... She would possesses control over her dreams, not quite to the aspect of a Child of the Underworld or Hyponos or Morpheous but far above average thanks to her Mother's Potent Mental poweress... as a rare power, her child may have control over fire that is beneath a child of Hephestus... Her children could enter any realm as Hestia is beloved by all... they would possess an immunity or at least resistance to fire... Also they may be able to absorb negativity, augment other people's powers through their fire and sacrifice ( Hestia is the Goddess of the Sacrifical flame...
She would more then likely hate violence and would do all she could to negate suffering. As a fatal flaw, she would either hold grudges ( Which Hestia holds towards her father and Zeus, hence why she has no kids) or would be an enabler.
More then likely those who desired Hestia would desire her...Perhaps a destiny tied to healing the rift caused by the Gods?
However since this is Cliche, I can better see a child blessed as her follower. Perhaps a clearsighted mortal, maybe a freind of Percy, being given a similar blessing to Hunters of Artemis ... maybe she becomes a Greek version of the Vestal Virgins? That seems more likely and more realistic...
Daughter of Hera
Born of Hera and a Spermbank supply? Or created like Pandora, with Hera using her blood to make her a child of her bloodline? As for why, perhaps during couples therapy ( A Canon thing) Hera's Therapist said to Walk a mile in her Husband's Shoes... so she created a child while stills staying faithful... inspired by her creation of Argus.
Regardless, this child would be born to suffer as Hera had countless enemies created from her own wrong doings. From the moment she enters the Greek world, she would be hated and assulted b just about everyone... Dionysius would hate her since Hera caused his Mother's Death, Zeus for all the crap she inflicted on his kids and lovers, Apollo for what Hera did to his Mother Leto. Hecate for what She did to her Daughter Lamia, and so many others...
Her Path would not be a fun one, she would more then likely be forced to abadon the Camp for her own safety though I think Hestia may protect her...who knows right? This story would focus on both the Daughter and Hera...I would insert her in the Third Book with Artemis offering her to become a Huntress before they learn of her bloodline...
Power- Potent Physical Strength surpassing children of Ares and coming close to Heracles as a Human since it was Hera's breast Milk that allowed Heracles to become so powerful. Empathy, mostly related to the Family Unit, as well as the ability to absorb tension. Potent Mental poweress, as Hera possesses these as well. As Hera is a rain goddess, she could make it rain and would have some control over the ground. She would have the ability to create curses, which Hera was famous for, but that would be situational and would require something to mimic the consequnces...
She may be able to create order and cleanliness, something Hera is known for, as well as cook incredibley well. As Hera is considered one of the Most intelligent and Beautiful Goddesses, this would translate into her Daughter making her Immensley intelligent, capable of creating clever plans ( As Hera did with Punishments or to end Conflicts like she did with Aphrodite). HEr Daughter would have beautify rivaling any child of Aphrodite...she would be almost immune to the Mist since, like her Mother, she could see the bigger picture however as a side effect she would be almost unable to manipulate the Mist...
Personality - She would have a great temper, Like her Mother but lack her mother's innate cruelty. Living a human life, she would be far less of a hypocrite and seek to help others as she feels responsible for them. She would be deeply intelligent and commanding by nature, much like the children of Zeus...Her fatal flaw could either be her temper, or perhaps her deisre to make up for her mother's monsterous deeds...
As for her Weapons... perhaps her Mother gives her a Lotus Staff, or something of that Nature... though, going with her physical strength I can see her using Celestial Bronze knuckles...
Side note, she would more then likely join the Titans for her own saftey... also Argus would have some connection to them through their Mother...
Daughter or Son of Eos- There are more then likely many, since she was cursed by Aphrodite...
They would be beautful with rosey skin and lustful dispostions... they would have the power to revive plant life as it was the Dew their mother brought that restored the plants. She would have power over light weaker then the children of Apollo. The Light of the Dawn would augament their powers, making them early risers. They may be able to make others more beautiful as Dawn is often considered the most beautiful time of day... they could heal others with the Dew of their mother at least to some degree...they would have a perfect sense of time... they have the ability to create new beginings, like their mother does with the creation of a new day. They would be immune to the Glare of Light... Grasshopers would be drawn to them as one of Eos's lovers was turned into a grasshoper by mistake... as the Herald of the Sun, Eos' children can augament and sense those tied to the Sun... As the Mother of the Wind, Eos' children have the ability to commune with those tied to the Winds and may be able to see Air Magic...Time magic would be less potetant against them as they are tied to the Passage of time through their Mother and sibligns ( the Horae)
Plot Points- Not sure to be honest, however what I can say is that their plot would start around Thalia's poisoning and may be able to slow the posions progress as Sunlight and Dew helps plant life...
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Children of Styx- Capable of entering the Underworld with ease, Capable of breathing underwater and controlling it to a lesser degree then a child of the Sea. They are immune to their Mother's River's acidic effect, can turn water acidic like their mother's River, can sense Hatred and Oaths... they may be blessed, to a lesser degree, with durability and battle proweress liek those that bathe in Styx's river... they would be able to enforce their mother's Oaths...They would have some control over Darker emotions, usually hatred, and be able to Shadow travel though it is more exhausting then it would be to a child of Hades...
They would be loyal, fiercely defensive of Oaths and hate those that break their word. They would bear a delicate, Nymph like presence and be feared by the living for they radiate the level of Hatred their Mother does...
( I really like the idea of one of these kids becoming freinds with Percy or Thalia or Jason since they are the children of Broken oaths...)
Daughter of Medusa- Born from a Blind Man and Medusa, the child is one of true Love and that was powerful. Medusa ended her own life to ensure her child could be born, as her body would no allow her to give birth. Her Child is immune to her powers, and is as beautiful as her old self was... She has a persuasive voice, immunity to Most poisons ( A Diluted ability from her Mother's Gorgon Blood) and is in fact posionus through her blood... She can command and commune with snakes, since she lacks Snake hair herself... her body is durable, like her Mothers. She can induce terror through a glare but it takes time to master... she is considered a Monster, but her Human side makes her a unique bred of creature... She can undo her Mother's Powers, restoring those turned into stone and her mere presence helps her mother overcome the insanity birthed by immesurable lonlienss...
Possible powers - The Child inherits the magic of her Divine Grandparents since Medusa was the children of Two Gods ( Keto and Phorcyus), granting her control over Sea Monsters, Sea Creatures and the ability to control water.
She has to be intelligent, as Medusa was a priestess of Athena and I cannot believe she would allow a dumb priestess in her midst
She may, at some point, grow powerful Nails like her Mother's Talons... there may be other traits from her mother like larger teeth ( Not tusks), enhanced strength and more...
She may be able to speak to Monsters unable to speak the Human tongue?
Path- She would be destined, destined to help her Mother but to ask as a bastion for those hurt by the gods. She would not seek to kill the gods, as they are their best choice for the time being, but to limit the damage... the Fates would protect her from outright destruction but she would still have an enemy in Athena...Posiden, out of Guilt, may help her in some ways and would let her sleep in his Cabin...
Techincially Kikyo and Kagome both died in the past seperate from Kagome's first lifetime... so it is possible that there are two versions of Kagome and Kikyo's Reincarnation born into the Modern World like Kagome! So random thought, some with those powers going back into the past...
To prevent an annoying paradox, the gods merge the souls and create a new soul out of it entirely... that soul could be reborn as a different being...
Reincarnation of Tsubaki- Blessed with power equal to a sealed Kagome, struggles with Inner Darkness and jealousy issues... Hates that everyone fawns over Kagome and becomes a rival of sorts for her...
Her Sould is damaged from the actiosn of her past life, so that plays some part in her life...
She dislikes Kagome, Miroku and Sango for their abuse towards Inuyasha and for their laziness... She would dedicate her time to breaking the spell on Inuyasha and convincing him that their so called freinds are assholes. They would develop either Sibling or Lover like bonds...
She would use a spear and be more practical, concious and intelligent then Kagome which would cause a lot of conflict. She cannot sense the Jewel...
Child of Sujin, the Lake Goddess- Breathes Underwater and controls Water... control over aquatic life... can purify and negate demonic and spiritual Powers...
Not sure of what else she would have, but still it is interesting...