Tough question...but thank you for asking.

It is very easy to get legalistic with an issue like this one and grace is still more important than anything else.  Your approach to family members must be delicate and considerate of age and spiritual maturity.  GK Chesterton once wrote, “…We have become so expert at that which is wrong, that the things that are good and right are misty and blurry to us.” [Paraphrase]  The difficult question to answer in the case of the ASU sundevil is whether or not “it” is a bad thing

The ideal situation is one where all family members agree to honor God in the same way with gladness.  Since the ASU sundevil poses no real threat to spiritual life, a significant amount of Bible study and prayer is in order before “making war” in the family over this issue.  [Some might say that one is could open the door to demonic oppression by entertaining icons such as these; but this is not true in the case of the ASU sundevil.  In order to have a real threat of demonic oppression, the object or icon must actually be a “dedicated” occult object whose purpose is worship of demonic power.  In spite of this, please consider the following verses:]

U       Ephesians 5:7: “do not be partakers with them.”

U       Philippians 4:8: “whatever is true…honorable…let your mind dwell on these…”

U       1 Corinthians chapters 8 – 10: a lot to say about idols and how it impacts relationships.

The ASU “sundevil” is in fact, a blatant trivialization of the devil, or perhaps some lesser demon.  A Christian should probably avoid any display of this icon because it sends a mixed message to the world, and a confusing one to new and/or weaker Christians.  An easy answer to the question of whether or not something like the ASU sundevil is appropriate in family life might be obtained by asking the following questions:

1. Would you wear the T-shirt, hat, etc. at church?
2. Would you be embarrassed by the display of it in your home when other Christians are around?
3. What is the personal intent behind the displaying of the icon?
4. Does the icon actually rate as an idol as the bible defines such things?
5. Are you willing to “die on this hill?”  In other words is the battle really worth it, or should energy be expended on other matters more crucial to family growth and unity.

Be sure you are placing the correct priority on this issue. It is not absolutely clear that there is anything wrong with the sun devil mascot. Good, conscientious  Christians would disagree over this issue. As such, it should not be considered of utmost importance to the point of offending and alienating friends and family. You need to recognize that unbelievers will not understand this; therefore, you should be careful how you handle or present the issue. 
 
If the ASU sundevil was as bad as say, Halloween, then the answer would be a clear and easy, “NO.”  Please note the emphasis above on might and probably before making a final decision and subsequent ultimatums relative to this issue.  Prayer, Bible study and grace are crucial to resolution of this problem; try to lead the family to unity as opposed to a decree of unity.