"Stay for awhile" is, technically, a grammar mistake. A preposition like "for" takes as its object a noun phrase — a noun or pronoun with or without modifiers. "For" is a preposition, but "awhile" isn't a noun. It's an adverb. So it can't be the object of "for."

"A while," on the other hand, is a noun phrase. It can be the object of the preposition "for." So "Stay for a while" is correct.

But if you take out the preposition, the dynamic changes, which is why both "Stay awhile" and "Stay a while" are correct. Here's a column I wrote a while back explaining why noun phrases like "a while" can function adverbially even though adverbs like "awhile" can't function as nouns.