This might seem a pretty straightforward word to spell, but there is the potential for confusion with a similar word role. The two words descend from a single Latin word rotulus, meaning a small wheel. The earliest uses of the word roll in English refer to the scrolls upon which medieval records were written. Because these scrolls could be wound up, like turning a wheel, the word roll was used to describe them. From this, the sense of a list or register developed, since these were written on scrolls of this kind. This is the origin of the phrase electoral roll, which should not therefore be spelled electoral role.
The spelling role is now used for a separate word which refers to an actor’s part in a play, or a person’s lot in life. This is a specialised sense of the word roll, adopted because actors’ parts were originally written on scrolls that could be rolled up. The earliest spellings of this word were role, rowle; the spelling rôle, modelled on the French spelling of the word, was first adopted in the eighteenth century. Given that role and roll are etymologically the same word, it would be possible to spell them both roll. But, given that actors no longer read from scrolls, people no longer see the connection; to spell the two words identically would be to lose a distinction that many today find useful.
But, should this word be spelled role or rôle? As long ago as 1926, Henry Fowler argued that, given that there was no separate word role from which rôle needed to be distinguished, the circumflex accent might as well be abandoned. But, while OED spells its headword as role, Chambers dictionary offers both alternatives. Discussion forums record lively debates concerning the correct spelling of the word, with some members reporting a tendency to prefer rôle when writing by hand and role when typing. For some contributors the accent is felt to give the word an ‘illustrious’ appearance, for others it lends the word an archaic feel; for some it is a way to look smart, while others consider it pretentious. Given the risks of appearing out of touch and affected, perhaps it’s safest to stick to role (but not roll)!