This is something that I've been wondering about today. I've only seen Habsburg but my phone is trying to tell me Hapsburg. How did the English language change this name and why is the autocorrect incorrect?
I've found only one question that asked this but he answer was poor and used Wikipedia as a source.
The definitive spelling of the surname is "Habsburg" and has been since at least the High Middle Ages. A number of surnames have undergone changes in orthography and spelling over time, but certain prominent surnames have generally remained the same. This is the case when referring to the House of Habsburg or, more formally, the House of Habsburg-Lothringen.
This article contains a seal and an early 15th century armorial painting of a cadet branch of the Habsburgs in Märstetter that uses the traditional spelling.
You just hit on something that I guess I kind of always wondered about, too. I'd really love to hear an answer. While we wait, I did an NGram search for English. It looks like "Hapsburg" was more common until around 1940, when it was overtaken by Habsburg.
Interestingly, usage of both was down considerably during WWII, and then rebounded significantly after about 1950, but "Habsburg" rebounded higher. This prompts some possible speculations, but alas that would be inappropriate. I wonder if anyone's ever written about it.
Here's a link to my Ngram seach, though I dunno if it will show up right.