He or she is not required to change his or her name unless it embodies the worship of someone or something other then God, Allaah. However, its amelioration by choosing a new Islamic name is legitimate and encouraged. The fact that he or she would change his or her name from a pagan or non-Islamic name to an Islamic one would be considered commendable--however, it is not mandatory.
Thus, if one's name is Abdul-Messiah, for example, or similar such names, then he is obligated to change it, as the Prophet (peace upon him) had people with the names Abdul-Ka'bah and Abdul-Uzzah change their names upon accepting Islaam.
If the original name does not comprise or imply anything forbidden in Islaam, then he or she is permitted to retain it (such as the name George, for example). As noted, though, it is preferable to change it to an Islaamic name, as this also distinguishes him or her from the kuffaar.
Note that if changing one's name in official documents and records poses a great inconvenience, it would suffice to change it among the people. In this case, he or she is called by his or her new name by friends, acquaintances, and the general public, while offical documents would retain the original given name.