Consider the sentence "She's a little crazy", taken from Disney's Aladdin. The copula verb has been attached to an adjective (the other common thing it combines with is an object), in this case the phrase "a little crazy". Since "crazy" is an adjective, "a little" is a qualifying adverbial phrase, somehow formed by combining the indefinite article with an adverb, whereas normally it combines with a noun to form a nominal phrase. (Of course "little" can be an adjective too, but I think it's an adverb here as it is in "little bit odd", where "odd" is an adjective and so "bit" and hence "little" is an adverb.)
As a follow-up, consider "It's a little bit funny", from Elton John's Your Song (lyrics by Bernie Taupin). Here "little bit" is an adverb, but so is "a little bit".
I'd be interested to understand the beyond-school-syllabus grammatical theory that makes sense of how these constructions are possible. How can "a" combine with an adverb to form an adverb? Or, if we see the structure as a (little X) instead of (a little) X, how can "a" combine with an adjective to form an adjective?