Oh this is fun.

I feel obligated to suggest Sticky McStickFace, but I don’t really like it.

You could run with a bitchy theme and call it Karen, “because it’s white and tells you where to go”.

Elsewhere you mentioned you liked Excalibur but were thinking of a female name: I could see Athena, who was the goddess of war and knowledge and the like. Or Miriam who in the Jewish tradition “ just as Moses led the men out of Egypt and taught them Torah, so too Miriam led the women”, which could work...

I looked up the (British) Queen’s scepter: one of them is “The Sovereign's Sceptre with Dove, which also has been known as the Rod of Equity and Mercy” so... Rodney?

Various directions there. Hope one strikes your fancy.

I’d go for a famous reference, like I named my hamster Splinter (TMNT) because he is a famous rat and my Roomba is Rosie (Jetsons) because she is a famous housekeeper robot...

Now we just need to think of a famous “stick.” Maybe a sword, broom, wand, or other item?

If you are a Game or Thrones fan there are some cool sword names: Needle (if you are a girl this could be especially fitting since it was Arya’s), Oathkeeper, Longclaw, Widow’s Wail, and Heartsbane.

OMG! naming a cane?? LOL.

m a u r i c e

Mine are all called Mister Pointy, after Kendra the Vampire Slayer's favourite stake. But I got rights on that. To their faces I tend to call them "You bloody thing" as the elastic breaks or goes saggy or whatever.

vastabat inclinata

Latin for swinging destroyer

I’m certain I’m not original - but King Arthur’s sword was called Excalibur and that would be a cool name for a cane.

I call mine light saber

Try something that starts with C, those names can be feminine but very pretty!
Candace, Camilla, Charlie, Cecilia, Callie, etc.

Or go for something that translates to its description,
In Greek, "white" is λευκό (pronounced- Lefko) but you could paint the name onto the cane to make it more personal and Greek is very pretty!

Choose a language you like, then a word you want to express, and find a feminine sounding name through it!

My fiance is a wrestling fan, so his is called Glenn (after Glenn Jacobs the wrestler whose character is called Kane)

Edward Worthington Sticksworth III. Or, Stick.

I call my walking stick

Sir Horrice the gummy of the black forest

It is a 200 year old walking stick head/handle is carved into a German dragons head but like me is missing half his teeth on one side.