There is a memorial service for Hagoita at Sensoji temple.History
It was used to be Omamori market held at the end of the year. It was called "Toshi no ichi, the market of the year." Today, they sale only Hagoita for Omamori.
Hagoita was Omamori for daughter because the feather of Hanetsuki, the badminton, looks like dragonfly which eat bugs. It represent keep the bad guy away from her. Also, the seed of the feather meant for keeps evil away and bring happiness. Therefore, it was very common to give Hagoita when people have daughter during Edo period.
Later, Hagoita became Omamori and one of the main products at Toshi no Ichi.