I call these things "Good Luck Cats" and pass by dozens of them everyday around Flushing, Queens. Just about every Chinese/Japanese business has one of these in the front of their stores/restaurants, as they are a symbol of fortune and are supposed to "beckon" customers to come inside. A raised left paw is supposed to attract new patrons, and a raised right paw is supposed to bring about money. Since new patrons = money, it seems irrelevant to me which paw the Good Luck Cat raises, no?
Once, while waiting for a table at a restaurant with my mom, brother, and cousin, my brother asked my mom about why businesses all have these cat figurines around and received that answer. This particular restaurant had an enormous ceramic Maneki Neko, and when my mom was explaining to my brother about the raised paw, he got confused about how a raised paw, palm out, was supposed to be a beckoning gesture, rather than a goodbye/waving away gesture. It seems that the Western gesture to beckon someone is hand pointed upwards, palm facing inwards, fingers waving towards themselves. I guess the Japanese/Chinese beckoning gesture is hand-up, fingers downwards, fingers waving towards oneself. Or something.
Personally, I do the whole-arm, all-the-way-out-to-my-right, back-to-my-body, one-handed-swimming-like motion. Or the diagonal all-the-way-downwards-to-the-right and bring-it-up-and-left motion, paired with an impatient "COME OOOOOOOOOOOON!!!!"