[roeme]

Interesting, I never encountered this phrase as well – is it dialect?

[dmm]

One of the definitions of stay is to stop something. So "I can't stay sleep." means "I can't stop sleep".

Shakespeare, Romeo and Juliet: V, iii:

"..But he which bore my letter, Friar John, Was stay'd by accident, and yesternight Return'd my letter back. "

[wavefunction]

It's just an awkward/antiquated phrasing. "Stay" isn't used much anymore except for "stay of execution."

[dllthomas]

Also, "stay one's hand". Still rare and a bit antiquated, but occurs more frequently than "stay" in this sense more generally (... is my impression).

For some data: https://books.google.com/ngrams/graph?content=stay+my+hand+%...

[disputin]

A stay of execution.

[LukeShu]

I (USA/Indiana) have mostly only encountered it in legal matters (recently: "Judge Young stayed today's order until ...").