A quick Google yielded no fruit; perhaps a language aficionado or expert could shed some more light.
Names ending on -er often refer to the person's origin (cf. "Ich bin ein Berliner"), and -ing or -ingen are typical endings of village or town names. So some ancestor could have been a Schrödinger in the sense of coming from Schröding or Schrödingen. There may have been several places of that name. As an example though, the wikipedia entry of a very old village named Schröding in Bavaria states that its name goes back to a man named Scort who lived around 800, and it was referred to as Scrotingin in a document from 1050. Place names on -ing or -ingen often go back to person names, so Scrotingin or Schröding(en) would mean something like "where Scort and his folks live". "Schröder" is also a common (much younger) family name that goes back to old words for taylor or drayman.
Of course, when refering to family name Schrödinger, these are all just speculations, common patterns that don't need to apply to this specific family. See also Wortherkunft.de, although they don't cite any sources either.
Erwin Schrödinger was an Austrian physicist, most known for his important contributions to quantum theory.
A popular meme is Schrödinger's cat, a thought experiment (1935) where a cat is in a superposed state of being both alive and dead at the same time if you use quantum mechanics to describe it. This paradox is still relevant when discussing the philosophy of quantum mechanics. Also, a fundamental equation in quantum mechanics is named after Schrödinger.