The character was describing her life, and has finished the previous sentence with a full stop. She then said・・・・・・なんてことない日々を変えたい、と焦がれながら. It's translated as "Hoping that some day something interesting would happen."
I am completely lost in the grammar of this sentence. I tried to read it as follows:
Xと焦がれながら = while longing for X.
But X is なんてことない日々を変えたい. So it's like saying "I was longing for me wanting to..." which clearly is wrong.
In addition, she's using a transitive verb 変えたい. Is she basically saying that she wanted to change the ordinary days, in the sense that she wanted to make something interesting happen?
And why 日々を変えたい rather than 日々が変えたい? Is it just conversational style?