While your name is one of your basic pieces of data, you are certainly allowed to change it. If you are changing your name when getting married (or divorced), you can do it for free; otherwise it costs HUF 10,000 for the first time (ca. EUR 25), and HUF 50,000 (ca. EUR 125) every subsequent time. If you have changed your name once, you can do it again only after 5 years (unless you are getting married or divorced).
You can choose any name that is in line with Hungarian naming traditions, fits your gender, and is not a historical name (the name of a historical family or figure, usually written with a traditional spelling). E.g. you cannot choose to be called Széchenyi or Kossuth; similarly, you cannot choose Eötvös, which is a historical family name, while you can go for Ötvös, which is a regular name (in fact, a name based on a profession).
When you change your name, you must change it everywhere: in your ID documents, with your bank, with your utility providers, and every other provider where you might be asked to identify yourself to access services. There is usually no strict deadline to do so, and the process is quite simple. However, you should take care of this as soon as possible so you can continue to access the services you are paying for without any hassle.