One of the senior Rabbis in Israel - Rav Chaim Kanievsky (son of the famous Steipler Gaon) - occasionally changes people's names.
This is often done when their Biblical names come from wicked people - like Nimrod.
A friend of mine - a scion of a Rabbinic family - was told to change his name from Amir to Meir. (The reason given by Rav Chaim was that Amir was animal food; though we have not been able to find a source for this.)
We see in the Chumash that Hashem renamed Abram to Abraham and Sarai to Sara and Yaakov to Yisrael - even though they were not ill.
Other biblical figures also had multiple names. For example Moses - as mention on Wikipedia:
Moses' other names were: Jekuthiel (by his mother), Heber (by his father), Jered (by Miriam), Avi Zanoah (by Aaron), Avi Gedor (by Kohath), Avi Soco (by his wet-nurse), Shemaiah ben Nethanel (by people of Israel). Moses is also attributed the names Toviah (as a first name), and Levi (as a family name) (Vayikra Rabbah 1:3), Heman, Mechoqeiq (lawgiver) and Ehl Gav Ish (Numbers 12:3).
Conclusion: There does not seem to be a Halachic issue with changing one's name, but I have not found any primary sources for this assertion.
That said, I assume there are Halachic implications when changing one's name - like when writing Gittin - a Get has to have the accurate name(s) of both spouses to be valid.