The problem with italki is that people answer questions when they don’t know the answer. This has nothing to do with “little” meaning “small”.

It means "have a small." It indicates that the plan is not significant or extensive. So, the sentence means that the speaker has a small or minor plan for the day.

I am an English language teacher. One of my students has asked the question below, please answer it: I have a little plan today. What's " have a little" It means "small" or " a small amount of?

"Little" turns "plan" into a diminutive. It has little to do with the size of the plan. It does not necessarily mean the plan is little. It only means that it is "little" in the speakers imagination. Saying "little plan" rather than "plan" softens the sentence "I have a plan". It makes the sentence more modest, and perhaps more polite.