As a German speaker and (I assume) English speaker, you're likely aware that the sound represented by ö is not at all common in English. One can argue that it's not like "r", but it's a closer approximation of the mouth position and sound than others. The IPA for Gödel is /ˈɡɜːrdəl/ : the symbols don't map one-to-one with English orthography (then again, what does), but it's no accident that there's an r in there. And then there's the question of whether or not to pronounce names closer to how they're pronounced in their native tongue, or how they're more properly understood, or some other rational.
Thanks for the reply! I didn't know the standardized IPA was in fact with an "r." Strange though, because to me the umlauted-o sound is probably one of the easiest German pronunciations to do for an English speaker. Also, good point on the compiled/interpreted (using a programming analogy) divide in pronunciation of borrowed words. I do suppose there is plenty of evidence of butchered French terms already, so why not ruin some other language's beautiful sounds too! groaning laughter
Though, I should not be all that surprised. For my entire life in the U.S. people have pronounced my last name of "Wiese" as either "Wise" or "Why-se," both of which are totally incorrect (my family says "Weese" when anglicized). I much prefer the German pronunciation of my last name, though; too bad it's too much of a pain to explain the discrepancy between how it's spelt and what it sounds like to native English speakers. :P