There was a time, before the age of the computer, when typographical systems had work-arounds where diacritical marks were not usually available. Ä, ä, Ö, ö, Ü, ü and ß were rendered, respectively, as Ae, ae, Oe, oe, Ue, ue and ss. I have never seen the name Goebbels rendered in any format other than as Goebbels, even in texts positively swarming with umlauts. Is it my imagination, or is there some reason that I have never seen the name as Göbbels?
Both versions are used if you have a look onto German wikipedia. Most likely you only have seen Goebbels, because it's the way the most "popular" Goebbels is written.
For names, the spelling is ambiguous - just because you speak an Umlaut, doesn't mean it's written.
Therefore you find people by the name of "Göbbels" and"Goebbels", but the same is true for the very standard "Müller" and "Mueller".
The spelling of phonetically similar last names often has a bunch of variations:
- Maier, Majer, Mayer, Meier, Mejer, Meyer.
- Schmid, Schmidt, Schmit, Schmitt, Schmied, Schmiedt, Schmiet, Schmiett
Why? Because these names are older than our "modern" definition of orthography....
Basically, the umlaut is a very reduced sign for the letter e. In „normal“ words, you would always write ö. But in names you could often find both, depending on the preference of the person at the time the orthography became fixed. (I can imagine that people called „Goebbels“ may change their name to „Göbbels“ to put in at least a little distance to the NS minister.)
Also, in some cases, esp. in middle and northern place names, e (and i) can indicate a long vowel. Examples: Coesfeld (pronounced Coosfeld), Voigt (pronounced Voogt).