“Let’s stay awhile.” Adverb modifying the verb “stay.”
“Let’s stay for a while.” Noun, object of the preposition “for.”
I imagine this distinction is destined for oblivion in relatively short order, considering how frequently these days I see constructions such as these, here on fluther and elsewhere:
“I’d like to be apart of the group.”
“That’s going to take along time.”
“Can you find away to help me?”