Arab culture had a widespread concept of nicknames of the form Abu X (Father of X) or Umm X (Mother of X). It is often after the first child, but it could be for other reasons as well.
Many people would be known more commonly by the nickname rather than their actual first name. Sometimes so much more that the first name becomes lost to history! Abu Bakr is an example of that.
This is the same reason someone would be named "son of Abu X." Take Abu Bakr's son as an example. Would it make more sense to call him Ibn Abi Bakr or Ibn First_Name when no one knows the First_Name that much?
The same applies to Umm Maktūm, his mother. She was commonly known as Umm Maktūm, hence that is how he was called: "Son of Umm Maktūm." The Arabs would see Umm Maktūm itself as a name/nickname by itself. So, they wouldn't look at it as "Son of the mother of Maktūm."