In-universe: We don´t know/Out of universe answer: to match the title of a symphonie
We don't know which culture Leela's family belong to or if all mutants use family names first.
However one Easter egg hidden in Futurama is the naming of Turanga Leela after the Turangalîla-Symphonie, a large-scale piece of orchestral music by Olivier Messiaen.
The author derived the title from two Sanskrit words, turanga and lîla, which roughly translate into English as "love song and hymn of joy, time, movement, rhythm, life, and death"and described the joy of Turangalîla as "superhuman, overflowing, dazzling and abandoned". This last description applies to some aspects of Leela's character.
(/speculation begins) The lîla Sanskrit word sounds much more like a given name, it seems the authors went with that and, to leave his name like the title of the symphonie, put her family name first. (/end speculation)