There are several synonyms for the phrase "become deeply lodged." Some of these include embedded, implanted, anchored, entrenched, and rooted. All of these words convey the idea of something being firmly and securely established in its place. When something becomes deeply lodged, it can be difficult to dislodge or remove. This could apply to physical objects like a splinter in your skin or a rock stuck in the ground, or it could refer to more abstract concepts like a belief or idea that is firmly ingrained in our minds. Regardless of the context, the phrase "become deeply lodged" suggests a level of permanence and stability.