I'm so tired, I can ........... stay awake.

'Hardly' is correct because it means 'scarcely' or 'barely', perfectly fitting the context of being too tired to remain awake. 'I can hardly stay awake' means staying awake is nearly impossible. 'Highly' (to a high degree), 'gradually' (slowly over time), 'honestly' (truthfully), and 'heartily' (enthusiastically) don't convey this meaning of barely being able to do something.