Final answer:
The Burbages converted an old monastery into an indoor theatre called The Black Friars Theatre, offering a space for performances during inclement weather.
Step-by-step explanation:
The indoor theatre that the Burbages converted from an old monastery during the time of the first Globe Theatre is known as c) The Black Friars Theatre.
This indoor playhouse provided a space for performances during the winter months when the weather was less accommodating for the open-air Globe.
The establishment of the Black Friars Theatre reflects the innovative spirit of the Elizabethan era, catering to the year-round demand for theatrical entertainment.