Actually, "to listen" is dinlemek. "to rest" is dinlenmek. They are close, but different.
As for the roots being related, I am not sure :)
It seems that tin, dinmek, dinlemek, dinlenmek are all ultimately related, at least according to Nişanyan's etymological dictionary.
He says that dinlemek (listen) is from old Turkish tıŋla- "kulak vermek, dinlemek" ("give ear", hear, listen) which he derives from tınıġ "nefes, soluk" (breath, soul) which in turn is from tın- "solumak" (to breathe). (Not sure where the leap from "breathe" to "hear" comes from....)
And dinlenmek (rest) he derives from tın/tıŋ "nefes, soluk" (soul, breath), with a note that "Tüm Türk dillerinde tıŋla-/diŋle- fiili "kulak vermek" anlamı kazanırken, sadece TTü diŋlen- başka bir anlama evrilmiştir." (In all Turkic languages, the verb *tıŋla/diğle-" gained the meaning "hear"; only Turkish-from-Turkey evolved a different a different meaning(?))
dinmek (abate, quieten, stop) is also from old Turkish tın- "1. soluk almak; 2. dinmek, dinlenmek" (1. breathe in; 2. quieten; rest). This makes a reasonable amount of sense to me - going from "breathe" to "stop to catch one's breath" to "rest" and "quieten".
And finally tin (soul, spirit) is straight from old Turkish tın "can, nefes" (soul). The initial t- is a bit suspicious as it should have been d- (din) if it had evolved normally; Nişanyan gives the note "TTü kullanılmayan bir kelime iken Dil Devrimi döneminde Kaşgarî'de bulunarak dolaşıma sokulan sözcüklerdendir. TTü biçimin normal ses evrimi çerçevesinde din olması gerekirdi." (While this is not a word that was used in Turkish-from-Turkey, it is one of the words that snuck in during the period of the Language Reform as (supposedly) having currency in Kaşgarî. The Turkish-from-Turkey form should have been din in the framework of the normal sound evolution.)
NOOO, mizinamo just explained it, dinleNmek means to rest, dinle mek(without the N) means to listen
But lucaturilli is right :) Dinlenmek also happens to be the passive form of dinlemek (to listen).
Ex: Bu şarkı arabada dinlenir. (This song is better listened to in the car.)
It is fine, but it can sound forced sometimes. I, an American, would rarely say "have a rest," although it is 100% ok. I added it as an alternative though.
(I always thought it sounded like a thing foreigners would say because it is in their textbooks.)
My translation is a korrekt translation. The English language use Gerunds as well.