Proper nouns are not covered by orthographic rules, full stop. That is the essence when asking for why a given person or a given place is spelt the way they are.
Excursion on the use of umlauts
For normal words, the general rule is to have ü for most /y/* sounds, ö for most /ø/ sounds and ä for a certain portion of the /ɛ/ sound. Neither of these rules is entirely strict, as /y/ can also be represented by y (Zylinder) and ö can also be represented by eu (Chaffeur) in words of foreign origin. The spelling of most words is fixed, i.e. *Zülinder is as wrong as *Blyte and *Chaufför (or *Schofför) is as wrong as *Geheur would be. (The former works better, because y can only be /ü/ or /i/ whereas eu is usually /oi/.) There are a few exceptions such as Friseur/Frisör in which two spelling variants are possible, but only for very common loanwords.
It is debated but possible that the umlauts derived from some shorthand form of e above a preceding vowel — in the Middle ages there was a plethora of these shorthands around, all of them unregulated, and since the Latin alphabet in itself does not provide adequate lettering for Germanic languages, it was a logical choice to make. Therefore, at one point it became acceptable to replace umlauts with the corresponding letter plus e. The pre-1901 German spelling rules actively enforced that partially by not allowing capital umlauts: It had to be Aepfel and Oel — a phenomenon still visible in proper nouns such as the town of Uelzen. The reasoning was maybe that umlauted capital blackletters looked less appealing. This was changed, however. Whenever possible, one should write the correct umlauts, capitalised or lowercase, and only if technical restrictions do not allow so is one allowed to resort to letter + e. (The only truely valid context I can think of is naming variables in programming languages that restrict names to ASCII characters — so
aepfelzahl would be acceptable there.)
Back to proper nouns
But the question was about proper nouns. Assuming a person could write, nobody could tell them how to properly spell their name. Not even the emperor tried to do that when accepting the 1901 spelling reform and the uproar would have been enourmous had they tried it in 1996. And remember that pre-1876, there were no official spelling rules at all. So people generally wrote freely — it has been mentioned that the well-known German poet and playwright spelt himself Goethe, Göte and Göthe on different occasions. Therefore, a lot of different spelling variants of identically pronunced names turned up, once literacy increased in the population. (And probably even before that, think christening lists in churches.) When finally people accepted a standard orthography for common nouns, there was a desire to also stick to the same way to write the corresponding common nouns. Since these were never regulated, many different variants still exist.
So in a nutshell, we write Gödel because Gödel’s ancestors used an ö in their name when it was fixed while we write Noether because Noether’s ancestors preferred oe. The surname Schmidt also exists as Schmied, Schmitt, Schmid and probably more although they all derive from Schmied. With Meiers it is even worse: each and every of the eight possible combinations is out there (Mayr, Mair, Meyr, Meir, Mayer, Maier, Meyer, Meier) sometimes in compounds sometimes not. Müller and Mueller are both common as is Hoffmann and Hofmann. I would imagine that apart from the reformator’s way of spelling (Luther) that Lutter and Luter should exist, too. You cannot even necessarily predict the pronunciation from the spelling: the novel of Thomas Mann is called Buddenbrooks although Buddenbrocks would have been a lot more logical in Lübeck, mainly because he didn’t want half the world to mispronunce their name (so urban legends say). A classmate of mine called Kroiher could have been spelt Kreuer by pronunciation, but a teacher once tried Kroher in analogy to Grevenbroich. And let us not forget the towns of Uelzen, Itzehoe and Buchloe, pronunced with an Ü, long o and o-e (two syllables), respectively.
And before you lean back please show me a language with not entirely phonetical spelling (i.e. Finnish doesn’t count) where you can immediately spell a person just from hearing their last name. Mr Davis/Davies? Mrs Brown/Browne? Mr Raymond Luxury Yacht?
*: I use the IPA symbols in a very generic way here, just to symbolise the sounds ä, ö and ü can make. I am well aware of the phonemic distinction between long and short vowels in German.