What argument could be made as to why the American colonies should not claim their independence from Great Britain?
The answer to this would have differed at different times in colonial history. For example, before the French and Indian War, one reason why the colonies should not have tried to claim independence was that they could have been attacked by the French in Canada. I will place my answer at a time in early 1775, after much conflict has arisen, but before actual combat has taken place.
At this point, one of the major reasons to have remained a colony was the fact that, as colonies, they were attached to the richest and most powerful country in the world. This was good for them economically and in terms of prestige. It would have been easy to argue that it made little sense to break away from such a situation.
A second argument is less pragmatic and is couched more in terms of what is right and wrong. From this point of view, the colonies were being selfish and unfair. They were resisting paying their fair share of the British taxes. They were doing things like destroying property (Boston Tea Party), which naturally made the British government angry. They were, in other words, not acting in good, law-abiding ways. Therefore, they had no right to be angry at the measures that the British were taking to try to control them.