Final answer:
The most important feature for this family when buying a new computer is the operating system, to ensure continuity with their familiar Macintosh environment.
Step-by-step explanation:
The most important feature of a new computer for a family that primarily needs internet access and does not play many games or watch movies would likely be the operating system. Since they are accustomed to a Macintosh, continuity with the Mac OS would provide a familiar user interface and continuity with existing software and files. While memory and processing speed are important, for basic internet usage, the requirements are not as high as for gaming or video editing, and the monitor size is a comfort factor that is typically not crucial for internet tasks.