How would parents affect their children's friendships if they
monitored their online activities

Explanation:

They often check the type of friends like they often advice their children to make good friends becoz if they have bad friends their children's attitude.

With the assumption that the underlined word in the sentence is ‘eagerness’, the answer that is synonymous to the word is letter A: enthusiasm. Enthusiasm, by definition is a strong feeling, in most cases, being excited, over something. So in the sentence:  

The Wanderer’s eagerness to work again was overwhelming.  

‘the Wanderer’s eagerness’, that is, the subject’s enthusiasm or excitement on to do something or to do a job, is so strong and he is clearly very determined as portrayed.