At no point are we told that it is Starbucks, and Starbucks has not paid any money for the promotion. We know that he likes his coffee Black, Strong, and possibly Jamaican Blend, but we are not given a brand. The cups look like Starbuick cups, but many coffee companies use similar cups.
I think its some sort of Jamaican blend or house blend. (Mark Harmon said in an interview a while back that most of the time, there is tea in his 'coffee' cups on screen; he only drinks one cup of coffee a day). :-]
everyone says Starbucks, but in fact it is not. If you actually look at the cup he carries around, it does not read Starbucks. It reads, "HOT, FRESH , COFFEE".. period. He does not reveal any brand of coffee in any show. He only reveals that he likes it Hot and Black. In Episode 21, Rekindled, he reveals that he doesn't have, make or drink coffee at home. He tells actor Jamie Lee Curtis, that he "Buys his coffee every morning". I guess we can assume that he likes Starbucks due to the similarity of the cup to a Starbucks cup, and perhaps it represents the black house blend. Other than this, it is all speculation. As an Army soldier, I am used to very hot, black and strong coffee, like at Starbucks. As a Marine, Gibbs would have experienced the same coffee, because the Army and Marines make it the same.