"Whatsoever things are honest" - Philippians 4:8
Biblical honesty is serious about a task. More often than not it refers to how a Christian is to live and deal with others. Grave is another word for it, and is used as a qualification of bishops and deacons. These men are required to take life seriously in order to fulfill their responsibilities adequately.
How much would your life change if you really took God and His Word seriously? If your burning desire was to obey Him and honor Him, keeping His commandments? Oh, how much our outreach efforts would intensify if we would take Jesus' Great Commission personally, and get to business on a daily basis. Heaven and hell are real, and it's very sobering to think about the eternal implications of choosing one over the other.
An honest person is very careful about his or her testimony. Knowing that we are ambassadors of God to the lost world, we desire to be the best example of a child of God that we can - not for the sake of our own salvation, but that we might effect the salvation of others.
That pattern of honest living is also critical when dealing with our brothers in Christ. People will give much more respect to a person who consistently and seriously deals with life and those around him. Don't get me wrong, sometimes the appropriate way to respond is with humor and lightheartedness. An honest person doesn't have to be boring, just not frivolous.