Specifically about food cravings, not greed in general. 嘴馋 means 'to have a greedy mouth / to be a foodie.' Often used playfully: 看着就让人馋 (just looking at it makes you drool).
Left food radical (side-form of 食) - the indexing radical. Pictograph of food piled on a serving stand. Anchors 馋 in the eating family with 饭 meal, 饿 hungry, 饱 full, 饮 drink. Marks the entry as an appetite/food-craving word.
Right side is a fused form of 兔 (rabbit) standing in for the phonetic of traditional 饞, which used 毚 - a two-rabbit hare graph. Simplification compressed that 17-stroke structure into a 9-stroke rabbit-shape on the right. Supplies the sound chán by inherited drift; no direct rabbit-meaning link.
