For the sentence "For this reason, the groups of small, decrepit houses were given his name", could another word for decrepit be run-down or old?

yes decripit means old, run down, dlapidated, etc...

Which one would be the best?

Both dilapidated and run-down are good words to describe a house.

Yes, another word that can be used interchangeably with "decrepit" in this sentence is "run-down." Both words describe something that is in a dilapidated or deteriorated condition. Additionally, the word "old" can also convey a similar meaning, although it is more general and may not convey the same level of deterioration as "decrepit" or "run-down." It is always helpful to consider the context of the sentence and the specific connotations of each word when choosing the most appropriate synonym.