What does one do when morality and honor conflicts with solidarity? Ideally, one would both be able to be honorable (i.e. to do the right thing especially with regard to others) and be in solidarity with relatives and friends (i.e. to support them, especially when they come into conflict with others).

What Biblical advice—according to Protestant interpretations and accepted texts, and a modern translation like NIV or ESV, if there is not a common view on this matter—is there for situations when these two virtues are at odds: when someone with whom you would normally be in solidarity does something dishonorable?  Does it make a difference how serious the offense is (e.g. allegations of abuse vs. going back on the spirit of but not the exact letter of an agreement about some mundane matter)?
Honor is not a particularly Christian virtue. It is mentioned biblically, but not nearly as often as other virtues we are expected to follow, particularly the most important one, Love. You haven't defined honor particularly clearly, but where your definition conflicts with Love,  a Christian should be loving rather than honorable.

As for solidarity with friends and relatives, the Bible (the New Testament) explicitly says  it is forbidden to favour your friends and relatives over other people. While there are certainly commands to honor those close to you ("Honor your father and mother") that does not imply that you should take their side just because they are related to you. "Honor" is not the same as the unthinking solidarity you seem to be talking about. If your father or mother do something terrible you are not expected to shield them from justice. Matthew 5:43-47:


  You have heard that it was said, ‘Love your neighbor and hate your enemy.’ But I tell you, love your enemies and pray for those who persecute you, that you may be children of your Father in heaven. He causes his sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous. If you love those who love you, what reward will you get? Are not even the tax collectors doing that? And if you greet only your own people, what are you doing more than others? Do not even pagans do that? Be perfect, therefore, as your heavenly Father is perfect.