Is it true that, based on his experiences and observations, Frankl argues that even when everything has been taken from you, it is possible to use what is left of your strength to retain some measure of free will by choosing how you think and respond?

Yes, Frankl makes this argument, developing it into his theory of logotherapy. In this theory, he argues that what drives humans is not primarily a death wish or a will to power but a desire to find meaning. We can retain free will by choosing to stay optimistic and by searching for meaning in whatever circumstances befall us.
As Frankl describes in Man's Search for Meaning, a terrible fate struck him: he was sent to Auschwitz. While he does not gloss over how horrible that experience was, and how much his survival was based simply on luck, he does also say that he chose to stay optimistic. He believes that attitude helped him, at least a tiny bit, to survive. For instance, he derived meaning and hope from thinking about being reunited with his wife after the war and pondering all that he would share with her. Even though she was dead at the time (which he did not know), his optimistic vision of a future with her helped reinforce his desire to live.
Frankl also notes that even in the most degraded and hellish of circumstances, there are still moments of joy. He recalls, for instance, a beautiful sunset the prisoners admired one evening. This is not to say by any means that life in a concentration camp was not a misery, but that we can chose how we respond to our circumstances in a way that catches what joy is available and embraces hope.