Well, for starters, Shakespeare used the word "honest" or some form of it a lot in Othello: over fifty times! So it's definitely worth our consideration.
First, the characters talk about who does or doesn't have an "honest face" or an "honest hand." In Act 2, Scene 3, Montano says it would be an "honest action" to tell Othello about how his second-in-command is an alcoholic. And there's a lot of talk about Desdemona and other women being "honest" or not. You get the idea that Shakespeare's use of the word "honest" in this sense is a stand-in for all kinds of virtues: like openness, faithfulness, and chastity. This makes sense if you take a look at the entry for "honest" in the Oxford English Dictionary and note that, in addition to "truthful," it also used to mean "noble" and "respectable" as well as "virtuous" and even "chaste."
Second, characters in this play sometimes call each other an "honest knave" or an "honest fool." You'll also find the phrase "honest plainness" in the play. In phrases like this, Shakespeare is using "honest" to mean "sincere" or "genuine." Again, the Oxford English Dictionary reveals that these, too, are valid definitions of the word "honest."
Third, and most importantly, many of the characters including the villain Iago himself constantly refer to him as "Honest Iago." In this case, Shakespeare is using the word ironically. It's funny; it's meant to entertain us: by continually referring to this villain as an honest guy, the characters are calling attention to Iago's deeply twisted, manipulative personality. And yet in some situations, Iago does often tell the truth--which only makes him more dangerous, since Othello doesn't know when he can trust Iago and when he can't.
The issue of Iago and his honesty becomes especially important in Act 3, Scene 3, when Iago is trying to get Othello to believe that his wife is unfaithful--and Iago calls attention to his own "honesty." He calls himself a fool for always telling the truth, which is ironic considering how he's currently lying about Othello's wife:
O wretched fool
That lov’st to make thine honesty a vice!
O monstrous world! Take note, take note, O world,
To be direct and honest is not safe.
Iago is seriously taking advantage of his own reputation as someone "honest" here: an incredibly dishonest thing to do.