A day will be 24h×60m×60s using a "mars second" equal to 1.0275 SI seconds.

'1.0275':14 '24h':5 '60m':6 '60s':7 'day':2 'equal':12 'mar':10 'second':11,16 'si':15 'use':8

A day will be 24h×60m×60s where a "second" is a unit of angle rather than time, equal to 1/86400 of a revolution w.r.t the sun (i.e. the equivalent of Earth's UT1, this is called "Coordinated Mars Time")

'1/86400':21 '24h':5 '60m':6 '60s':7 'angl':15 'call':37 'coordin':38 'day':2 'earth':32 'equal':19 'equival':30 'i.e':28 'mar':39 'rather':16 'revolut':24 'second':10 'sun':27 'time':18,40 'unit':13 'ut1':34 'w.r.t':25

A day will be 24 hours, 39 minutes and 35.244 SI seconds long

'24':5 '35.244':10 '39':7 'day':2 'hour':6 'long':13 'minut':8 'second':12 'si':11

Time of day will be an incrementing number of SI seconds from 0 to 88775.244 (possibly rounded to an integer, with leap-seconds every 4 days or whatever, or no leap seconds and allowing it to drift)

'0':13 '4':26 '88775.244':15 'allow':35 'day':3,27 'drift':38 'everi':25 'increment':7 'integ':20 'leap':23,32 'leap-second':22 'number':8 'possibl':16 'round':17 'second':11,24,33 'si':10 'time':1 'whatev':29

A day will be 24h×60m×60s using SI seconds (and will drift with respect to mean solar time by ~40 minutes a day)

'24h':5 '40':21 '60m':6 '60s':7 'day':2,24 'drift':13 'mean':17 'minut':22 'respect':15 'second':10 'si':9 'solar':18 'time':19 'use':8

Mars has a solar day equal to 24 hours, 39 minutes and 35.244 seconds. This is 2.75% longer than an Earth solar day, and close enough to the period of the human sleep–wake cycle that it's plausible civil timekeeping on Mars might be based around it.

Once a (official or de facto) standard  concept of civil timekeeping exists on Mars, what will be the first standard way to talk about the time of day among ordinary Martians? This market ignores questions about how a Martian calendar will otherwise be set up, it's exclusively about how Martians will talk about the time of day within a solar cycle. It also ignores the concept of timezones - resolution of any of the answers doesn't depend on whether everyone on Mars uses the same timezone or separate ones.

This requires there to be a concept of ordinary civilians who live on Mars and aren't there as employees of an Earth-based mission - we are asking about what system ordinary Martians will use for timekeeping day-to-day.

There may be multiple standards used in different jurisdictions on Mars, or among different groups of people. Whilst I'll try to resolve based on whichever seems to be the dominant one when such standards first emerge, if it's close I may wait to see if one reaches dominance. The spirit of the question is to resolve based on what most people actually use day to day, whether it's an official standard or not.

There are a few candidates that come to mind:

This will resolve YES if the solar day is divided into 24h of 60m of 60s each, where a second is a fixed interval of time defined with respect to fundamental constants such as atomic transitions. This Martian second need not be 

 as 1.0275 SI seconds - it's fine if Mars defines their own standard with respect to a different atomic transition or whatever, as long as the result is a martian second that is 1.0275 SI seconds long. Mars need not use this same "Martian second" in engineering or physics, this question is only about civil timekeeping.

This answer will resolve YES regardless of whether leap seconds are used to keep the time system in sync with mean solar time (as opposed to just allowing for drift).

A day will be 24h×60m×60s where a "second" is a unit of angle rather than time, equal to 1/86400 of a revolution (i.e. the equivalent of Earth's UT1)

This is a system where a "second" isn't a unit of time at all, but a unit of angle of rotation of the planet with respect to the sun. The time duration corresponding to a second isn't fixed, and watches and clocks need to sync with astronomical observations of Mars' rotation on long timescales in order to know what time it is is this time system, even if they're as precise as atomic clocks. However, this system needs no leap seconds to stay in sync with mean solar time - it 

 mean solar time, essentially being constantly kept in sync.

People will keep using SI seconds, and just deal with the fact that the day is not a whole number of hours long, and the last hour doesn't have 60 minutes, and the last minute doesn't have 60 seconds. Watches will tick over from 24:39:35 to 00:00:00.

Again, this answer resolves YES regardless of whether leap seconds are used to keep this time system in sync with mean solar time.

Not too much to explain about this one. They'll count SI seconds. Maybe they'll keep the day to an integer number of seconds, and maybe they'll correct for the resulting error with leap seconds, or maybe they'll just allow for the drift compared to mean solar time.

That is, they just use Earth days, and tolerate that sunrise and sunset move by 40 minutes a day.

Feel free to recommend more, including systems not even approximately based around the Mars or Earth solar cycle. But we're looking for a description of how regular Martians will talk about time on a timescale comparable to or less than a Martian solar day, for example how they will schedule appointments a few hours out and communicate to each other when to meet.

This question's close date should be extended until Mars colonisation happens, and then long enough for de facto standards to be established among ordinary Martians.