You are quite right to put "honesty" in quotation marks, for any mention of Iago as being honest must be ironic. However, the irony is mostly dramatic. All of the major characters believe that Iago is honest, so when the term is used, the audience knows that each of these characters has misjudged Iago.
One of the first references to honesty is made by Iago himself, in the last few lines of Act 1. He characterizes Othello as one who
is of a free and open nature
That thinks men honest that but seem to be so
A perceptive judge of human nature, Iago understands that Othello is truly honest, and because he is so, he sees others as he is himself. Thus, Iago knows that Othello will trust him.
Later, Cassio calls Iago "honest" after Iago gives Cassio advice that will result in Othello's ordering Cassio's death:
Good night, honest Iago.
But the term reaches its highest irony when Othello calls Iago honest:
And, for I know thou'rt full of love and honesty
And weigh'st thy words before thou giv'st them breath.
And later, Othello declares,
This fellow's of exceeding honesty,
And knows all qualities, with a learned spirit
Of human dealings. (Act 3, scene 3)
These references to Iago's honesty, while initially are somewhat humorous because the audience knows how grossly Othello has misjudged Iago and his intentions, are the key to understanding why Othello falls for Iago's machinations and turns against his most beloved wife Desdemona. It is Othello's faith in those around him, his tendency to see others as he himself is, his trust in the men who have served with him in battle that leads to Othello's downfall. In other words, much of what causes Othello to turn against Desdemona is his inability to see the darker nature of Iago.