Paul Schoendorf, my great-great-grandfather, settled in Cleveland, Ohio at the end of the 19th century. He came from a village outside Gersheim, which is now in the German state of Saarland, near the border with France. The name Schoendorf has a number of features that are very typical of German names. Words and names are often long and complex in German. However, by breaking words into their parts it can be easy to discover the meaning of a name. In turn, this my give clues to the origins of your family.
The name Schoendorf can be easily broken into two parts. The first part of the name comes from the adjective schön, meaning "beautiful". The second part of the name is the German word Dorf, which means "village". It is a common element in many German place names, perhaps most famously in the city of Düsseldorf. And, because it is common in place names, it has also become common in surnames, as in the name of the founder of Bergdorf Goodman department store in New York. Together, the meaning of Schoendorf is the somewhat generic "beautiful village".
You may have noticed that the word for beautiful in German has the letter O with two dots written above it (ö). This character is usually referred to as an "O with umlaut". The name Schoendorf would also be written with this symbol in a German-speaking country. So, when looking in German databases, it may be necessary to type the name as "Schöndorf". When the umlaut character is unavailable, instead of simply dropping the two dots, the letter is written with a following E. This is also a very common convention that has been followed by many German immigrants when adapting names. For example, the very common German name of Müller is often found as Mueller. If you have a name, which you think is German, that has vowels "oe", "ue" and "ae", you may want to consider substituting these vowel pairs for the vowel with umlaut, "ö", "ü" and "ä". You may find something new this way. If you are going to type these characters a lot, you may want to look into the keyboards settings on your computer. If not, it is easy to produce these symbols in Microsoft Word. Simply type ctrl + shift + ; and then type the desired vowel. You can then copy and paste where needed.