The sentence I want to parse is from page 6, chapter 1 of がばいばあちゃん. It reads:
I get the general sense of the sentence. It's saying you can liberate yourself from that kind of life. The って replaces という. The last part, I guess, could be glossed as, 'That kind of thing [i.e. liberating yourself] has got to be possible'. But I'm not sure how the 思うことだって... works exactly. My guess is it comes from 思うことできる (It's possible to think) and the って highlights 思うこと, so maybe you could translate it as,
Such a thing as thinking there is a new chance to liberate yourself from such a life - it's got to be possible!
However, if that were the correct reading, I'm not sure what the だ in the だって is doing exactly. If anyone could help me out, it would be much appreciated.