You can get fairly complicated with this one.

We had a thread not too long ago about Mars time vs Earth time--there are a couple of possibilities--either have Mars time units (hours, minutes, seconds) be the same length as the earth time units and have an odd number of time units in a day, or adjust the length of the Mars units so they come out even (in which case a Mars second won't be the same length as the Earth second). There are arguments in favor of either approach. Another possibility is to try to resurrect the old French decimalized clock and calendar, but adapted for Martian time. I don't recall if the French concocted new names for their new intervals or not--if they did, you might be able to mangle them into Martian names. Or you could name the time units after, say, names concocted for the old Battlestar Gallactica TV series.

Once you figure out the number of Mars days in a Mars year, you might try to work out a calendar. Most likely there won't be an even number of days--you might want to figure out a leap year system. Isaac Asimov wrote an essay quite a while back on a revised earth calendar--he proposed tossing out months altogether and having a calendar consisting of 4 Seasons--each Season would have 91 days (conveniently a multiple of 7) with 1.25 special days a year. These special days wouldn't have a weekday associated with them--Saturday, the 91st of Winter would be followed by New Years Day which would be followed by Sunday the 1st of Spring. Every four years or so, Saturday the 91st of Summer would be followed by Leap Day which would be followed by Sunday the 1st of Fall. The advantage of this calendar is that you need only one--the 35th will always be on a Saturday, regardless of whether it's Spring, Summer, Fall, or WInter.

You might see if you can work out something similar with the Martian year.

If you want to get really tricky, you can look up the equation of time, and try to work out Martian Daylight Savings Time.