"'s" indicates possesive case. Although it's absurd to say that "today" owns "weather", possesive case can indicate other relationships too. For example, in "Picasso's paintings" it means "by Picasso" and in "children's stories" it means "for children". So "today's weather" can be interpreted as "weather for today". So all is good.
BUT there's another seemingly valid construction, namely "I want to know today weather", where "today" is a noun. We could say that in said phrase "today" is an attributive noun (A.K.A. a noun adjunct), thus implying that it means "weather for today". (Like in "dog food" word "dog" is an attributive noun that makes meaning "food for dogs". In other words, an attributive noun can indicate "for".)
Is there anything wrong with my reasoning? I was told that "Today weather" is incorrect.