As I've noted before, in modern storytelling, characterization is a primary value. When evaluating a novel, readers and critics often make their judgment based on how dynamic and well-rounded its characters seem: a book that gives us a realistic and introspective portrayal of its characters is considered of literary value, while stories that more interested in plot or action are seen as second class.
Abraham's flavor of doubt probably isn't the same as what we moderns might experience. It's unlikely that he ever seriously questions the existence of God or the possibility of miracles like we often do today.
This cultural climate poses a challenge for adaptions focused on the life of Abraham and his descendants. Although Genesis does provide subtle bits of characterization to those who read the dialogue closely and carefully track the use of keywords and other literary devices, it's safe to say that the narrative is not interested in giving readers the kind of psychologically introspective portrayal that most moderns expect. In order for the story of Abraham's family to meet modern expectations, the subtle hints of characterization embedded in the biblical narrative need to be made more explicit and accessible, and characters who are given a fairly flat portrayal need to be reconstructed with more depth and development.
When Heroes of the Faith Doubt
In Jewish and Christian tradition, Abraham is often upheld as a hero of the faith because of how Genesis says God credited his faith as righteousness and because of his willingness to offer his son, Isaac, to God. And yet Abraham's character in Genesis is far from a shining example of unwavering faith. On multiple occasions, he gives his wife away to other men in order to save his own skin, and he also famously takes Hagar, his slave, as a concubine in an attempt to conceive. Abraham's son, Isaac, and his grandson, Jacob, follow in the same path, sometimes taking incredible steps of trust in God and at other times acting with a surprising degree of faithlessness.
The presence of these notable successes and failures make it clear that a core aspect of the character of Abraham and his descendants is their struggle to believe in the face of doubt. What Genesis doesn't make explicit is how that struggle actually plays out internally.
Gods are notoriously fickle and difficult to please, and so it would be natural for Abraham to question whether his patron has had a change of heart.
We can surmise that Abraham's age and the barrenness of his wife make it difficult for him to trust that Yahweh would fulfill his promise. But Abraham's flavor of doubt probably isn't the same as what we moderns might experience. It's unlikely that he ever seriously questions the existence of God or the possibility of miracles like we often do today. If Abraham's doubt is metaphysical, he's most likely questioning whether Yahweh specifically has the power to carry out such a miracle. An adaptation could dramatize this kind of struggle by showing Abraham getting tempted to turn aside to other deities when it seems like Yahweh isn't getting the job done.
It may be even more likely, however, that Abraham's struggle is less conceptual and more relational. Instead of doubting whether Yahweh has the power to carry out his promise, he's probably questioning whether Yahweh has the willingness to do so. Gods are notoriously fickle and difficult to please, and so it would be natural for Abraham to question whether his patron has had a change of heart. To dramatize this kind of struggle, a Job-like dialogue (perhaps between Abram & Sarai) might be necessary, because the opposing perspective is a little less clear cut.
More Voice for the Voiceless
Another challenge in adapting the Patriarchal narratives is that they are, as one might expect, patriarchal. Our focus lies on the actions and ambitions of the male characters. To be fair, Genesis is actually quite egalitarian relative to other stories of its time. It takes a surprising degree of interest in the struggles of women, particularly oppressed or downtrodden women like Hagar. Relative to modern expectations, however, the female characters of Genesis lack complexity and really only exist to further a male-centered plot.
Of course, there already are attempts to retell the Patriarchal stories from a feminine perspective, most notably, Anita Diamant's The Red Tent and the subsequent adaptation. But an adaptation need not attempt to completely reorient the story of the patriarchs in order to give more voice and complexity to the women involved. At the very least, an adaptation can strive to pass the Bechdel test by constructing female characters whose concerns extend beyond finding a husband and having a son. Genesis understandably focuses on both of these concerns because its plot is focused on the Promised Seed, but there are other plot or thematic functions (e.g. as we suggested above).
The Pagan Mind
The original audience of Genesis was submerged in a pagan world. They didn't need to be told what kinds of things the characters who interacted with Abraham believed or the religious practices that they engaged in. Modern audiences, on the other hand, are quite distant from the worldview of the Ancient Near East. As a result, it's far more difficult for us to conceptualize the "bad guys" or even the "neutrals" that come into the story - because Genesis makes little effort to fill in the gaps that we bring.
Any adaptation of Genesis should work a bit to help us understand the mindset and practices of the pagan characters of the Patriarchal world. This is particularly important for the "bad guys," because it can help us understand where they're actually coming from and make them into more interesting foils for Abraham's character.