What I am going to suggest is that you don't waste your time researching family stories with regards to a change of surname if you have no back up of it other than family stories...... just write them down, who told you and when, and that is important as in a few months time the story you know now will be different.....it is very scary when you look back 6 months later and read a story you wrote as it will not be the same details you remember and if anyone else tells you, you write their version down too...........
Surnames are just an identifier for all of us and in the past and especially in that part of Europe a surname identified not only you and your family, but your religion and political persuasion.........so could get you killed.....and depending on the era as some era's were more dangerous than others.
Follow the paper trail of records your family left behind and if there is a name change you will find it, it comes down to proving it is the same people, however often that is not too hard to do ( depending on year and records available)
Look at the history of the countries at the time frame you are looking at, when civil registration came in and there is a list of 'Global Civil Registration' on the 'document' page of http://familytimeline.webs.com/ click the 'Other record resources for family history' on the drop down menu if you are not sure when that was, but that helps you by knowing what records could be found, so if you know civil registration wasn't set up then it is parish baptisms you would look for and if it was then you know you have civil births and parish baptisms.....and that makes it easier to cross reference information.
Unless you are in Hungary, make sure you get onto all the forums for it and not just the American ones, they ar e good BUT they are too big and it can take years if ever before anyone gets back to you...most European countries have their own forums and the obvious ones to go to are the british ones...same language, good research skills, daily active...so you will get answered the same day..links page on the same link I gave you Britgen, rootschat,...can't think what others but it is under forums section......also the surname boards same place..... the UK also have a very large Hungarian community, so you may come across lots of genealogists on the UK boards with the right background to help you..................