After naming our son Rahim we have been told that it is prohibited to say Allah's Names like that. We renamed him Abdul Rahim, but on his birth certificate we still have only Rahim. Should we change it? Is there any sin?
Rahim is not one of the names of God. Al-Rahim, on the other hand, is. Calling your child's name thus, with the definite article Al-, would be problematic; in such cases you would appropriately go with Abdul-Rahim.
Without the definite article, Rahim is just an adjective, and one not unique to God (As Najeeb describes in his own answer, it's used by God Himself to refer to the messenger). As long as it's an attribute that is not unique to God (e.g. Muhyi or Mumit), using it as a name is fine.
See also the answers to this related question: Is naming a person using one of Allah's names allowed?
I don't think there's anything wrong in calling a human Rahim or even Rauf since the Messenger (SAWS) himself has been referred to as such in the Qur'an:
"There has certainly come to you a Messenger from among yourselves. Grievous to him is what you suffer; [he is] concerned over you and to the believers is kind [
Rauf] and merciful [
Rahim]." [Qur'an 9:128]
(Latter two interpolations mine.)
Rahim is not the same as Al-Rahim .. and you certainly do not have any intention to match the attributes of God that Al-Rahim was so InsyaAllah it would not be a sin. But it's better that he meant the name Abdul Rahim servants of the Merciful God ..