Should an invite say at the Smith home or at the Smith's home?

At the Smith home is good, but at the home of Janet and Robert Smith (or whatever the couple's name is) might be even better, since it's more personal and conversational. As for the other choice, it is not correct: at the Smith's home is a singular possessive, meaning there is only one Smith. For a plural possessive, you need Smiths'-- at the Smiths' home.