The story
By haplologic alteration of umbilic- + -ectomy (which reflects the surface analysis of the word). The non-haplologic form would not present any pronunciation difficulties (it would be /ʌmˌbɪlɪˈsɛktoʊmi/), but the fact that the haplologic form both was coined and also was disseminated as the normal form of the word is a testament to the idea that haplology is driven by various cognitive processes besides phonemic elision.
Etymology from Wiktionary, CC BY-SA 4.0.