(I was going to put this in Mariology, but I figured it'd get more response as a separate thread.)
In Paper Mario, we get to see Bowser's diary. The entries are all from the month of ☆, and the days of the entries (which may or may not be consecutive, but are definitely in order) are as follows:
O/O
O/X
X/X
In Thousand-Year Door, we see several entries from Flavio's log. They are again in the month of ☆, but the days are:
XO
XX
X♪
X☆
X♥
OO
OX
O♪
O☆
O
In TTYD, it seems to be a base-6 number system. Oddly, X comes before O when it's in the tens digit, but comes after it in the ones digit, and when there are slashes (I don't recall there being any days with slashes in TTYD, so it's unclear whether they would come before or after the numbers without slashes. I'm currently replaying it and hoping I'll find some later on.).
There is little room for skipped days in Flavio's log entries -- The first three entries are explicitly shown to be consecutive days by the rising and setting of the sun, and the third one, Day X♪, is on the day they were shipwrecked. He then writes three entries, Days X☆, X♥, and OO, and on Day OX writes "Three days on the isle now." O♪ and O☆ are when Mario enters and exits Cortez's cave, respectively (I doubt Mario spent more than two full days in there, if even that. He probably went in at evening and came out the next morning or afternoon.), and O is after it's all over (possibly skipping a day (as no Day O♥ is recorded) to give him time to collect his thoughts, though it could also be a typo or just weirdness in the way Mushroom Worlders count).
Since XO through OO are consecutive, it seems rather clear that O represents 1 and ☆ represents 4, and so, X-O discrepancy aside, it is likely safe to assume that ☆ is the fourth month of the year.
Having established that ☆ is the fourth month, it remains to be determined how long a month is on the Mushroom Planet. Presumably it is based on the phases of the moon, but the moon in TTYD seems to be much smaller than Earth's, as Mario is able to traverse it in about six screens (though the gravity is more similar to our moon, which makes sense in conjunction with the gravity seen in Galaxy. It's also possible that Mario wasn't traversing the whole moon across its widest point, but rather a latitudinal circle near a pole). There is also at least one more moon, the very small one in Yoshi's Island; this one is probably too small to be seen, but it's possible that it's depicted smaller than it actually is due to the art style.
If we can figure out the month and day system, it could then be combined with the year system in Super Paper Mario, based on the founding of the Flora Kingdom. Granted, the Flora Kingdom is in another dimension, but apparently in Mario's multiverse, alternate dimensions run concurrently in time, similar to the way time travel works in Partners in Time, where it's more like there's a Present World and a Past World that flow simultaneously and can interact, so I say run with it. Super Paper Mario takes place in Flora Year 1626. Thousand-Year Door gives the ages of several characters, including Toadsworth, so if we can figure out how long afterward SPM takes place, we can get approximate birth years for them.
(The symbol numbers seem to only be used for dates; most everything else seems to be on a standard Arabic numeral base-10 system.)