It is difficult to answer this question directly since it asks whether "here" in the example sentence is an adverb or an adjective. As tchrist remarks, this word does not easily fit into either category.
If, however, one rephrases this question in more general terms, it suddenly becomes a very worthwhile topic of reflection. Consider it phrased this way: "In the sentence 'This book here is the one I was talking about' , what part of speech does 'here' fit?"
Now, it is clearly not an adjective for the following reason. An adjective says something about what a noun is, in addition to whatever the noun itself says. (I am, however, interested in FumbleFingers' comment about the way the word is treated dialectically.) In the phrase "this book here" , 'here' , doesn't tell you anything about what the book is. It does, though, tell you something about the book's way of being, namely, where it is. So we can say that 'here' serves to locate the subject noun of the sentence ('this book').
But it serves as a locator in a way that needs to be treated on its own. Compare the difference between these two examples: 1.) "This book here is the one I was talking about." 2.) "This book in the store is the one I was talking about." 'In the store' is an objective locator that could be pointed to on a map. 'Here' , on the other hand, is a subjective locator since it depends on the proximity of the speaker for its meaning. 'Here' , due to its dependence on the speaker's location, belongs to a grammatical category distinct from prepositional phrases such as 'in the store...' or 'on the floor...' Words of this category have traditionally been labelled "deictic" (from a Greek adjective meaning "that which displays, that which demonstrates") . Deixis is an important rhetorical trope that helps a speaker bring the attention of his audience to bear on a single, tangible point.
Based on this analysis, I would call 'here' in the above example a deictic adverb of the locative variety ( Temporal deictic adverbs do exist, and I want to distinguish the use in question from such a temporal use.). It is an adverb because it describes not what it is but how it is . It is deictic because it depends for its meaning on a proximity between the speaker and that to which the speaker referred. And, finally, it is locative since it refers to where the book is located.
The complexity of the grammatical analysis required to establish this simple fact is a testament to the shadowy area of semantics and syntax we have stepped into when we encounter deixis. Is a deictic word really a word, with a definitive grammatical part of speech if it depends for its meaning on non-linguistic objects? Have we not discovered that bugaboo of grammarians everywhere, context?