What can a motif contribute to or become when it gathers larger significance throughout a text?

The short answer to your question is the theme. Motifs, like many literary devices, contribute ultimately to the theme or the overall message of the book. Motifs are really just patterns in literature as they are anywhere else - and any pattern in a work of literature has some significance. Think of motifs as a stone rolling downhill, picking up things along the way. It gets large and larger. As such, the more a motif is mentioned, the more significance it picks up.

An example of a motif from a famous work of literature: rain or water in John Steinbeck's The Grapes of Wrath. You don't really need much knowledge of the book to immediately make a connection between water and grapes. Ya need water to grow grapes, yes? It's ultimately, life-giving, torrential water that fuels the rage of the oppressed and grows the so-called grapes of wrath. Water comes up a lot in the book. In a way, it's central because the characters are facing a drought. This has symbolic meaning, that is, both literal and figurative. Anytime a motif has symbolic meaning, you can bet it probably contributes to the theme. Theme, as its used colloquially, is interchangeable with motif. You often hear people speaking of "recurring themes" throughout a novel. Well, there's really no such thing as a recurring theme in a novel unless you're speaking of a theme recurring in individual plots or subplots of a novel. Theme should be a comprehensive ultimate message or conclusion, however - so it's not really recurring. What IS recurring is the motif, which contributes to the theme. So death, life, water, sun, air, fire, often Words like these, these create motifs or patterns, and when you piece together a pattern, making it holistic, you get a theme.