What is the ruling on eating meat that was offered to us, when we do not know whether its source was stolen or given willingly?

The fact that your neighbor works in a royal palace does not necessarily mean he stole the meat. Raising possibilities of theft or illicit giving constitutes ill-suspicion (su'u az-zann) towards him. The default rule for things is permissibility (ibahah), and the Muslim's earning is lawful. Therefore, there is no blame upon you for eating the meat, unless you have a strong presumption, based on powerful evidence, that it was stolen.