"If you want to make this a business partnership, Mr. Hart, with all nonsense cut out of it, I'm in on it. I know something about vaudeville teams in general; but this would have to be one in particular. I want you to know that I'm on the stage for what I can cart away from it every pay-day in a little manila envelope with nicotine stains on it, where the cashier has licked the flap. It's kind of a hobby of mine to want to cravenette myself for plenty of rainy days in the future. I want you to know just how I am. I don't know what an all-night restaurant looks like; I drink only weak tea; I never spoke to a man at a stage entrance in my life, and I've got money in five savings banks."

"Cravenette" sounds like the name of a fabric. But what does
she mean by it in this context?

Signals over. I had a hunch and searched for "Cravenette cotton wool" and found this:
"CRAVENETTE. Reg. T.M. of Cravenette Corp. for a water repellent process for fabrics."

the word lingered in NY till the early 1950's, (i would have said its a kind of water-resistant head scarf (of decorative cloth)

by then most women of a certain age would use rain bonnets that they could keep folded in their purses, but i did hear cravenette used as a term for a head scarf. ( i always thought it was some sort of version of cravate. (a term for a scarf worn as tie by a man)

i always thought it was some sort of version of cravate

Cravat, meaning tie, comes from the French adjective for Croatian.

Cravenette, according to this website:

http://www.rtwear.com/glossary/c.htm

is a registered TM of the eponymous company. And according to this:

http://www.sdc.org.uk/pdf_features/JA99F2.PDF

they used to be a British company.