Bondying is indeed a proper noun, the name of a fictional character. However, this character has been around for decades that reference to him reminds people of certain traits that do not need to be explained. This would be similar to incorporating names like Superman, Batman or Spiderman in a sentence. Most people would have some idea who you are talking about and the archetypes represented by these characters without having to go through a lenghthy explanation. Using proper nouns as adjectives, adapted verbs or as an embodiment of an ideal or trait is common in English (Don't be such a Scrooge. My brother's a real Einstein. Pasteurized milk. Every artist dreams of finding his Maecenas).
The character Bondying is essentially a big baby, with the exaggerated traits of a big baby. As with proper names in English that are used in ways other than to pertain to a specific person, Bondying can be used similarly. Those familiar with the character don't need an explanation when they hear the word pabondying-bondying (lazing around or loafing), a verb adapted from the proper noun Bondying or when they hear the sentence "Para kang Bondying" (depending on the context it can mean You look like a baby or You're acting like a big baby).
Having said all that, I still say that momonding is too idiosycratic that it would be almost impossible to derive a meaningful linguistic source. It is not a Tagalog word, though that does not mean the originator did not modify a Tagalog source (let's say Bondying). I am not aware of any Tagalog lullaby that uses the word momonding. It is possible that it is based on another dialect. More likely it is typical of original lullabies that parents or grandparents sometimes create spontaneously, incorporating made-up words that are meaningful only to themselves.
Since the originator is deceased, a clue may lie in the dialect she habitually spoke or the name of your brother (could momonding be a rhyme for a name like Ramon or Dindo?). Also though your brother may have been very young at the time, sometimes sensory impressions persist. Perhaps he knows what momonding means. His opinion would be more valid than mine.
