For example, these characters have different radicals:
根 跟 恨 狠
But the part of the characters minus the radical is the same.
Does that part (艮 in this case) have a name?
On wiki, this part is defined as "phonetic". I am not familiar with these jargons in English, but I thinks this one is acceptable.
In your case,"艮"(gèn or gěn) is really a character.
Edit: Add Chinese names for this part
We call these characters “形声字”(in modern Chinese, many characters can be classified into this type). 形 means "form", referring to the radical, 声 means "sound", so I think "phonetic" is acceptable to call that part.
In Chinese, we call the "radical" 形旁 and the "phonetic" 声旁. Here, 旁(pang2) means a part.
Victor Mair, the well known sinologist (and occasionally outspoken advocate of pinyin) consistently uses the word phonophore to describe the phonetic part of the character.
If you start saying "phonophore" too, then there will be upwards of two people using the term. :-)