What may have been the motivation for Shakespeare making a major character in Othello a black moor when the actors and audience in Elizabethan London would have most likely been all white?

The short answer is: for the drama. Shakespeare was adapting a previously-told story, the Italian author Cinthio's "Un Capitano Moro", and he took significant liberties with the story's characters and plot -- the skeleton of the narrative remains, but the story in its original form was unlikely to be familiar to Shakespeare's English audience. (This is pretty typical for Early Modern English drama and for Shakespeare; Shakespeare also took the skeletal plot of Measure For Measure from another tale by Cinthio.) The Moorish captain of the original story is still a Moorish captain, so Shakespeare wasn't imposing a racial element on the story out of nowhere -- another question might be "what motivated Cinthio to make the title character of his tale a Moor?" (Besides his pretense that he's simply retelling a true story that was told to him by the story's Emilia-analogue.) Cinthio's Moor isn't persecuted by his ensign on account of his race or on account of some unknown malice, but because his Iago-equivalent wants to sleep with Desdemona and is rebuffed by her; the Moor's race comes most into prominence with the matter of Desdemona's handkerchief, embroidered "in Moorish fashion", and with the Signori of Venice's outrage not only at Desdemona's murder but Desdemona's murder at the hands of a barbarian. Is this supposed to be a cautionary tale about not marrying outside your own race because Moors are jealous and will beat you to death with a sock full of sand (seriously) or is it a cautionary tale about not believing the lies of hunky but depraved ensigns? It's a little bit of a mixed message; the racial element is unmistakably present, but not as elaborated-upon as Shakespeare's play has it.

Shakespeare took that racial element and made it even more explicit, so that it informs every character's relationship with Othello either overtly or covertly. He invented the character of Brabantio to combine the narrative roles of Desdemona's kinsmen and to an extent the presence of the Venetian signoria -- but also to give Iago somebody to interfere with to create trouble for Othello, and as a concrete representative of racial prejudice. Desdemona's parents urge her to marry someone else in Cinthio's story, but they don't use the same language of racial opprobrium to do it in, nor is the scandalous element so passionately emphasized. Shakespeare also injected a topical political element by emphasizing Othello's role in a religious conflict -- Othello isn't white, and as a Moor he might be understood as essentially non-Christian (either pagan Muslim or as Iago implies to Brabantio, out-and-out satanic) but he is allied with white, Christian Venice in the ongoing struggle against Turkish forces. This positions him as sympathetic and familiar to a Christian audience and makes him plausibly attractive to Desdemona, but it still leaves him on thin ice. Othello's strength and intelligence have been applied in the defense of Venice against the Turk, but his social acceptance in Venice is conditional, and his Moorishness sets him apart as never completely belonging there, even as a war hero with Venetian superiors, Venetian underlings, and a Venetian wife. The conflict between Christian and Turk wouldn't have been lost on Cinthio writing in Italy, but Shakespeare brings it to the forefront in a way that's absent in "Un Capitano Moro". In Shakespeare's play, Othello even compares himself in the act of suicide to a "malignant and turbanned Turk" he once slew in Aleppo -- the Turkish conflict is everywhere, as ubiquitous as the theme of race, and it situates the human drama of Othello's rise and fall in the context of a political conflict that held a place of concern in English audience's hearts not only for its religious implications but for its humanitarian concerns. So you've got perpetually topical political conflict built in, military themes, familiar and crowd-pleasing themes of Italian impetuousness and prejudice, and the outline of a plot that incorporates romance, pathos, suspense, and tragedy, plus a passel of roles for the King's Men to take on: a really loathsome and chatty villain, a zesty flawed hero for Richard Burbage to play, an obnoxious old man, a clever maid, a beautiful and virtuous heroine who gets to lie around being unfairly maligned and tearjerking without having to give too many long speeches. It's a little sexy, a little challenging, but not too challenging. Race isn't one of the thematic elements that run the risk of a little visit from law enforcement, either, so why not make a play about the Moorish captain?

It's also possible Othello was written as a cash-in on a theatrical vogue for nonwhite protagonists, one that didn't leave much concrete evidence behind besides the existence of other plays with racially marginalized antiheroes and secondary black characters, and it's possible it was written as a more sympathetic and sophisticated return to the theme of interracial union and racial prejudice he'd explored maybe a decade earlier with Aaron the Moor in Titus Andronicus. We have records of black people living and working in England and participating in the same business activities and social obligations as their white counterparts, so a black person living in London and choosing to pay for a show and watching a performance of Othello isn't beyond the realm of possibility. But outside speculation, we don't know how they'd have felt about the action of the play itself -- seeing a tragedy about race and sex play out with a famous white guy pretending to be Othello? Seeing a dark-skinned tragic hero whose nobility of character is undermined by other characters' racial prejudice? Seeing an exotic black hero in an Italian tragedy and not a likeable black hero in a domestic comedy? Seeing a stage representation of a black man kiss and kill a stage representation of a white woman? Black people who saw or read Othello in its earliest years of performance must have had thoughts and feelings about it, but we have no written record of those thoughts and feelings.

The limitations of the Elizabethan stage necessitated a willingness to actors playing what they weren't -- of course this is relatively innocuous when it's a bricklayer's son playing a Danish king or a Venetian beauty and somewhat less so when it's a white gentile reproducing a mean-spirited caricature of a Jewish person or a black person. Jewish characters were depicted on the English stage with relative frequency despite there being few or no people of Jewish descent in the average English audience and no actual Jewish people on the stage -- real-life "blackamoors" would have been more familiar to more of Shakespeare's audience than real-life Jewish people. In 1603 England you could run into a black person you knew on the street right outside the Globe Theatre, or hire a black servant, or marry a black person in your parish church, or attend the christening of a biracial baby there. Jewish people of any complexion, even ardent converts to Christianity, would have been excluded from nearly all of these things and had been excluded by law for centuries; even high-profile Jewish conversos like Roderigo Lopez were held in suspicion. So why write a play with contemporary Jewish people in it at all? The difference implied in Jewishness for a character like Shylock in Merchant of Venice or Barabas in Jew of Malta still meant something to Elizabethan audiences -- that difference signified an existential and political threat, a hostility toward Christianity and a certain parcel of villainous qualities. Shakespeare writing Jewish people in Continental Europe doesn't cleave exclusively to the territory of well-worn stereotypes about Jewish malice and greed, but neither does he deliberately flout or divorce himself from contemporary stereotypes wholesale; Jewish people were marked off as foreign, grudgingly tolerated at best, and that foreignness set them apart on the stage. Likewise Othello, being not just a London "blackamoor" but a Venetian Moor, is essentially different and essentially foreign -- not just in the way that Italian characters on the stage differ from English spectators in the audience but different even among fictional foreigners -- but as a presumed Christian, he can make a somewhat more comfortable tragic hero rather than simply an antihero. This situates the actions of such characters at a comfortable remove, rendering them exotic and dramatic rather than familiar. For that matter, the fact that the other white characters in the play are not English but Italian also situates them in an exotic, somewhat stereotyped ethnic space -- Shakespeare's plays set in Italy like Romeo & Juliet and Measure For Measure and the plays of Shakespeare's contemporaries get significant traction out of contemporary English beliefs about the essential nature of Italians as passionate, temperamental, lascivious, and furthermore Catholic. Perhaps these plays got more thematic leeway as a result than they'd receive from audiences if their source-texts were transposed to familiar English surroundings. Shakespeare's Othello is firmly situated in the realm of foreignness and difference, even with its white characters, and that may have motivated Shakespeare to go further off the beaten track with Othello than with his other tragic protagonists.