This rare kanji has no clear standalone meaning. Treat its meaning as uncertain.
棰 combines 木 (tree/wood) with 垂, which likely contributes sound. The exact historical development is uncertain.
A wooden (木) object hanging down (垂) like a stick or cane. The shape suggests something made of wood that hangs or is used to strike.
For ソイ, imagine a soy-sauce-soaked wooden stick: soy -> ソイ, and the wooden stick (棰) is used to stir the sauce.