Both the Jacksonians and the anti-Jacksonians were trying to claim the heritage of the Republican Party of Jefferson. (Calling Jefferson's party the Democratic-Republican Party is about as historically accurate as referring to the post-Justinian Roman Empire as the Byzantine Empire.) The Anti-Jacksonians tried to broaden the party to be more than a faction of former Republicans by changing the name. Associating themselves with the name Whig (which historically was an anti-monarchial name in its various incarnations) also served to indicate they thought Jackson was acting like a king instead of a president. So the Whig Party was opposed to King Jackson.
That also sowed the seeds of their downfall:
I really like this book on the subject.
About the same time that the name was adopted, was when you had people come over to the party who were "State's Rights" Whigs like White and Tyler, who disagreed with the core of "National Republicans" who largely agreed on support for tariffs, bank etc.
Another interesting dynamic was the influx of Anti-Masonic people into the Party, as well.
