The sol is divided into 24 major units (Martian equivalent of hours), which are in turn divided into 60 smaller units lasting 61.64947506 SI seconds (based on a value of 88775.24409 seconds for the sol). Expressed in Terran units, the "Martian hour" lasts 61.64947506 minutes, or 1 hour 1 min and 38.9685036 seconds. Clocks that display only Martian hours and minutes look exactly like Terran clock. Clocks that display seconds use the format 24:60:62 (i.e. display resets to zero at midnight after reaching 23:59:61).
Pro
Given that the second is almost never used in social and business schedules, and that many clocks do not display seconds anyway, the system will feel identical to the terrestrial clocks to most users.
No leap seconds or timeslips are needed. Time is counted continuously throughout the sol. Time-based calculations and unit conversions can be done essentially as on Earth, except that the factor 61.64947506 is used instead of 60 when converting between Martian “minutes” and seconds.
Keeping the SI second means that other units and constants derived from it (Hz, G, c, definition of the meter etc…) are the same on Mars and Earth. Physics textbooks and computer software don't need extensive rewriting on Mars.
The same method, i.e. stretching or shrinking the minute to express it in a decimal number of SI seconds) can be used to generate clock systems that use the second on other planets or satellites, not just Mars.
Con
Having to remember and use the number 61.64947506.