What does Mitt Romney aide's remark "keep Boston off message all week" mean?
"Off message" means:
not adhering to or reflecting the official line of a political party, government, or other organization
Boston is used here to mean Romney headquarters. It is the object of the sentence.
"Keep Boston off-message" means "Prevent the Romney campaign from making its own points". Presumably, this news has the potential to make the Romney campaign defend itself on this off-message point for a while to explain an irrelevant or deleterious narrative, instead of staying on-message and telling its own narrative.