In "The Last Leaf," why does Sue call Johnsy "white mouse"?

The term "white mouse" has two parts, "white" and "mouse." Sue is visually oriented because she is an artist. She sees Johnsy as a white mouse for two reasons, both of which can be identified in O. Henry's text.

“Tell me as soon as you have finished,” said Johnsy, closing her eyes, and lying white and still as a fallen statue...

Johnsy is as white as a marble statue because she is so sick. Sue...

The term "white mouse" has two parts, "white" and "mouse." Sue is visually oriented because she is an artist. She sees Johnsy as a white mouse for two reasons, both of which can be identified in O. Henry's text.

“Tell me as soon as you have finished,” said Johnsy, closing her eyes, and lying white and still as a fallen statue...

Johnsy is as white as a marble statue because she is so sick. Sue naturally notices this paleness, and it explains the "white" in "white mouse." Sue is trying to be amusing to get some spark of interest from her patient. It is noteworthy that Sue talks to Johnsy in little jokes and even baby talk but that Johnsy does not respond in kind. Johnsy is appropriately taciturn, solemn and serious as befits a person who is dying. Then when Johnsy is on her way to recovery she uses one word of the intimate language the two girls have evolved during their time as roommates. Johnsy says:

“I've been a bad girl, Sudie,”

Sue has been using jokes and baby talk throughout the story because she has come to regard Johnsy as her sick child. Johnsy may have brought out Sue's maternal instinct. She says:

"Try to take some broth now, and let Sudie go back to her drawing, so she can sell the editor man with it, and buy port wine for her sick child, and pork chops for her greedy self.”

Getting back to the word "mouse," O. Henry tells us that Johnsy is a very small young woman.

A mite of a little woman with blood thinned by California zephyrs was hardly fair game for the red-fisted, short-breathed old duffer [i.e., Mr. Pneumonia]. 

So Sue jokingly calls Johnsy a "white mouse" because pneumonia has made the mite of a little woman deathly white. The whiteness probably scares Sue, but she is trying to keep up a brave front whenever she is with her "sick child," which is something most parents will do when they are nursing a little boy or girl who is seriously ill.