The word notice has to do with paying attention to something. For example, you usually notice a flashing light, and an overdue bill might come with the word "NOTICE" written on it to make sure you see it.
The Latin nōtitia, a noun meaning "something known," was the ancestor of notice. Use the verb to mean “to see or point out" and the noun to mean "formal warning." When you give “advance notice,” you let someone know early or ahead of time. In case you haven't noticed, the notice posted on your door says you've been promoted and should move to the corner office. No doubt because of your attention to detail.