Why is the servant-candle on the menorah called “shamash” (meaning servant) instead of “eved” which is the word for servant in Hebrew
"Shamash" is related to the word "shimush" meaning "use" or "serving". The shamash candle is "serving" the other Chanukah candles by lighting them. Similar to a waiter who "serves" the food. While not in use in common Hebrew, I have heard of a maid being called a "shamash". Likewise, the shul's "caretaker" is called the "shamash" because he serves or provides the needs of the shul. Often, he does this voluntarily including "mundane" tasks such as sweeping the floors and turning on the lights and heat before people came in to daven. (Used to be more common decades ago than it is now. Seems that now, we pay NON-Jews to do this job! Sad, in a way.)
One explanation I heard from someone a while ago is that "shamash" is spelled the same way as "shemesh" meaning "sun". In the case of Chanukah candles, like the sun, the shamash provides the light for the other candles. Of course, this is stretching the explanation, a bit, but, then again, it's possible that the word "shemesh" used for "sun" may have originated from the word "shamash", anyway.
Also, slight nuance. "Eved" means "worker" or "slave", implying that there is a boss or master or someone controlling the eved. "Shamash" doesn't imply that there is a boss, necessarily (it could, though). The shamash candle has no "boss". As a matter of fact, it IS the boss!