Without the word "about", the sentence is simple:
It weighs a pound.
That's a pronoun, a verb, and a noun phrase acting as a subject, a transitive verb, and a direct object.
Even with the word "about", much of the grammar of the sentence remains the same:
It weighs about a pound.
"It" remains the subject, "weighs" remains a transitive verb, and the phrase "about a pound" might still be the direct object.
I think we can immediate discount the possibilities that this "about" is a noun, verb, conjunction or interjection. We only need to examine the preposition, the adjective and the adverb as reasonable possibilities.
The word "about" is often a preposition, and "about a pound" could be a prepositional phrase. If it is, we should be able to find other prepositional phrases which fit this sentence pattern.
We learned early American history.
We learned about the pilgrims.
I cleaned the floor.
I cleaned under the bed.
There is a clear general pattern here, and there are at least two ways to explain it. One obvious explanation is that prepositional phrases can be direct objects. Another explanation is that prepositional phrases can modify the intransitive use of a normally transitive verb. In either case, the pattern Subject / Verb / Prepositional Phrase make sense.
This is reasonable evidence for the "about" in question being a preposition.
If it is an ordinary adjective, it would appear after the definite article: "an about pound". If it modifies "a pound" at all, it must be a predeterminer. Unfortunately, I don't see how to distinguish between this possibility and the possibility that it is a preposition.
Instead, I'll rely on Oxford Dictionaries. That source lists only preposition and adverb as possible grammatical categories. In contrast, its listing for all does include the category predeterminer.
Either Oxford Dictionaries is mistaken, or the word "about" is not a predeterminer.
If it is an adverb, then it must modify the verb. There is no other word in this sentence that an adverb could modify. If so, we should be free to change its placement:
It about weighs a pound. (x)
It weighs about a pound.
It weighs a pound about. (x)
It only weighs a pound.
It weighs only a pound.
It weighs a pound only.
This is sufficient reason to claim that the "about" in question is not an adverb.
In the sentence "It weighs about a pound", "about" is a preposition and "a pound" is that preposition's object.