Most names have some nickname form, e.g.,
- Júlia - Juli
- Erzsébet - Erzsi
- Pál - Pali
- Péter - Peti
There are some that are not that apparent (András - Bandi, István - Pista or Pisti, etc.). For maximum effect you can also apply the endearing suffix (idk what the proper term is), -ka/-ke, like: Julika, Petike, Pistike but never "Pistaka". This can also be applied to other things like animals (like tehén (cow) - tehénke) but in that case it can also just mean "little". But I don't know any funny like Richard - Dick.
Actually my examples were not that good because Júlia can also become Julcsi and Erzsébet Bözsi... It's complicated... I found this list if you are interested: http://corpus.nytud.hu/utonevportal/html/bec.html