I don't know the exact adventures of Bugsy Eyes as I'm not familiar with this particular bedtime story character. It could be that Bugsy Eyes goes on a journey to find a lost treasure.
I'm not sure specifically who 'Bugsy Eyes' from bedtime stories is as there could be many different bedtime stories with unique characters. It might be a very specific character from a particular book or a local folktale.
Not having read the story, Bugsy could be a variety of things. Maybe Bugsy is a small insect with a big personality. He might be the kind of character that children can easily relate to, someone who has little adventures during the day and then comes home to a warm and cozy bed at night in the story.
I don't have enough information to determine who Bugsy is. It could be a fictional character created specifically for this 'bugsy out of bedtime stories' concept.
It's possible that Bugsy is a wise character. Despite being part of bedtime stories, he could be the one who imparts valuable lessons. For instance, he might teach the other characters (and the children listening to the story) about the importance of sharing, or how to be brave in the face of fear. He could be an old, experienced creature in the fictional world of these stories.
Bugsy might be a unique character in the bedtime story. He could be a brave little being. For example, if it's a story set in a forest, Bugsy could be a squirrel who is known for his daring adventures. He might be the one who always explores the farthest parts of the forest, looking for the biggest acorns or the most beautiful flowers. Or he could be a friendly bug that helps other forest creatures when they are in trouble.