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Make it so the player can give Asterion certain accessories to wear. 
The player gets these items by meeting certain conditions in the game. 
Some of the accessories are tasteful, simple, and even personally meaningful. 
Others may be less so. 
Some of these accessories may cause dialogue to occur or actions or scenes to become available depending on how the meaning of the accessory worn relates to the situation at hand. I've left the details vague on purpose; it is up to the authors to decide which accessories are meaningful or not/ which ones make certain dialogue, options, and scenes possible or not. 

Possible accessories include:
Agal, Armband, Armring, Bracelet, Brooch, Cestus, Cameo, Collar, Cow Blinders, Graphic Pin Button, Cumberbund (goes with tuxedo), Eye Patch, Monocle, Necklace (so many possibilities, sometimes the most simple object strung around the neck can be the most meaningful), Nosegay/bouquet, Paper Bag w/ Eye Holes, Pin (could be helpful if Asterion has any long hairstyles), Reimiro (a crescent-shaped pectoral ornament once worn by the people of Easter Island), Rings, Sunglasses (Shutter Shades only available on "I speedrun games" playthrough), Wreaths for the head (olive, bay laurel, thorns, gold-wrought olive), Wristwatch. 
https://en.wikipedia.org/wiki/Jewellery
This may provide some inspiration, they go into detail on what kind of jewelry ancient people had.

Also, if you play the game in such a way to make Asterion not just serve you but really care for/love you, this results in him giving the player such an item of significance.