Why does 息 mean news? Why does 息 mean news?

The chraracter's etymology suggest it means "rest" - 
"From nose indicating breath and also phonetic 自 and heart or chest 心. Meaning to rest."

Source: http://chineseetymology.org/CharacterEtymology.aspx?submitButton1=Etymology&amp;characterInput=%E6%81%AF

So I guess, it is something to do with nose + heart as a metaphor for "interest". It might be understood as "What is dear(heart) to me(nose)"
Over its long history of usage, the meaning of 息 has evolved.

Yes, 息 has the connotation of message.


  (5) 消息 [message]
  (6) 又如:信息(音信;消息);息耗(消息,信息)


Its original meaning is to breathe;pant.


  (会意兼形声。从心,从自,自亦声。自,鼻子。古人以为气是从心里通过鼻子呼吸的。本义:喘气;呼吸)


And then the meaning was extended,


  人之氣急曰喘。舒曰息。引伸爲休息之偁。又引伸爲生長之偁。引伸之義行而鼻息之義廢矣。


[ Breathing slowly was called 息, and it extended again to mean rest, and extended again to mean grow. ]

And then it became to be used with its antonym 消 (get rid of;eliminate) as 消息 from pre-Qin period (先秦), means 增减;盛衰 (growth and decline), and then extended to mean 变化 (change).

Then 消息 began to be used to mean 情况 (circumstances;situation;condition).


  到了魏晋时期,“消息”的涵义则多为“情况”了.
  任何事物都会有从生长到死灭、从兴盛到衰败的发展变化过程,这种变化过程是一种“情况”.所以“消息”从“消长(生死盛衰)”的涵义演变为“情况”的涵义。


[ By the Northern and Southern Dynasties, 消息's primary meaning was situation. The process of evolution is a kind of situation, so 消息's meaning was extended from changing to situation. ]


  到了南北朝后期,“消息”便又发展出“音信”、“音讯”的涵义,而且从此就成为“消息”最主要的意思.


[ By the later stages of the Northern and Southern Dynasties, 消息's meaning was further extended to mean 音信 (mail;message;news) and 音讯.  And that became its primary meaning.]

Reference:
释息
“消息”的词义演变历程探析
从消息到信息──词义的变迁与汉语教学隅例