そんな人生から開放される新たなチャンスだと思うことだって、できるはずだ。(『がばいばあちゃん』page 6, chapter 1)
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.
Maybe I hit upon the answer (or an important clue)? The main grammar point used is ことできる (is possible). The だって probably means 'even'. So perhaps the sentence could be parsed: "Even thinking there is a new chance of being liberated from such a life - that's got to be possible." In the next sentence the narrator says something about couples and families talking among themselves about what kind of effort they should make into the future and how this will get rid of the communication gap.