 Solution

Continuity gives existence of a minimizer. If there were two, choose a rational $q$ strictly between them. Then the minima on $[0,q]$ and $[q,1]$ would coincide. Conditional on ${\mathcal{F}}_q$, the latter equals $B_q$ plus the minimum of an independent Brownian motion on $[0,1-q]$, whose distribution is continuous by the Brownian reflection principle. Thus equality has conditional probability zero. Taking the countable union over rational $q$ proves almost-sure uniqueness.