How can I name my days of the week?

The modern weekdays are actually not named directly after gods, but after planets. This system started with the Babylonians, who named each day (*) after one of the seven classical planets: Mercury, Venus, Mars, Jupiter, Saturn, the Moon, and the Sun. This system was borrowed by the Egyptians, then the Greeks, then the Romans, then various Germanic peoples, using local equivalents to the Babylonian names each time, which is why we now have "Sunday" and "Monday" (from "sun" and "moon") and "Saturday" (from Saturn, who had no Germanic equivalent) alongside an eclectic assortment of Germanic deities.

If astrology is important in your conculture, a similar system would make sense. You could name the days of the week after planets, constellations, important stars, or any heavenly objects that are important to these people.

(*) Strictly speaking, they assigned each hour to a planet, then named each day after the planet that governed its first hour. Cassius Dio has a more elaborate explanation.

Following all the restrictions given in the question, I propose using letters of the writing system as names for days of the week.

If your society is literate and the number of symbols in the writing system is relatively small, the canonical order of letters is going to be very well known.

I don't know of any natural languages that actually do this, although the heavenly stems were historically used to label days of the week based on which sacrifices and rituals were supposed to be performed when. So, the heavenly stems form a small closed system with additional meaning besides designating a day of the week.

Tying the symbols-as-days-of-week to a religion, current or historical, is probably a good way to make the system plausible in your conworld.

---

So, you specifically mentioned not wanting to name days after numbers, but some very widely spoken languages do exactly this.

I think you can make the 12-day week in two halves thing make sense by having a duodecimal number system with six as a sub-base. In a system like this, some or all the numbers in the 7 to 12 range would be derived from the words for 1 to 6.

In Mandarin Chinese, the days of the week are simply numbered, with Sunday being an exception to the pattern.

• 星期一 (week 1) means Monday.
• 星期二 (week 2) means Tuesday.
• 星期日 (week sun) or 星期天 (week sky) means Sunday.

In addition, some number systems have a sub-base.

For instance, here is a numeral system developed for an Inuit language with a vigesimal number system by speakers of the language. The number system has a sub-base of 5, which is reflected in the forms of the numerals.

Here is the table in the article reproduced below. The system is not completely regular, but many numbers in the 1-20 range, particularly those congruent to 2 or 3 mod 5 are transparently derived from a small multiple of 5 followed by the word for 2 or 3. Some numerals have irregular forms or are represented subtractively like 9, 14, 19.

0         1                2                   3                    4
-         atausiq          malġuk              piŋasut              sisamat

5         6                7                   8                    9
tallimat  itchaksrat       tallimat malġuk     tallimat piŋasut     quliŋuġutaiḷaq

10        11               12                  13                   14
qulit     qulit atausiq    qulit malġuk        qulit piŋasut        akimiaġutaiḷaq

15        16               17                  18                   19
akimiaq   akimiaq atausiq  akimiaq malġuk      akimiaq piŋasut      iñuiññaŋŋutaiḷaq

20
iñuiññaq