day, sun, Japan, counter for days
日 centers on the sun and the day it defines: the bright celestial body, the period of daylight, and by extension the concept of a full day. The Japan sense is a cultural application of the sun symbol, and the counter for days follows naturally from the day meaning.
day; days
Sunday
Japan
today; this day
tomorrow
every day; daily
yesterday
Japanese person
Japanese (language)
hello; good day; good afternoon
everyday; daily; ordinary; usual; routine; regular
today; this day
day in question; appointed day; that very day; the day (of issue, publication, etc.)
daily; everyday
weekday; ordinary days (i.e. non-holiday)
diary; journal
holiday; day off
the other day; a few days ago
Sunday
Sunday
previous day; day before; eve; prior day; preceding day
next day
Saturday
morning sun
日 is a pictograph of the sun, originally a circle with a dot or line inside. The modern squared form with a central horizontal stroke is a stylized version of that ancient sun image.
The rectangular outline is the sun's disk, and the horizontal stroke inside is its bright center. See 日 as a simple drawing of the sun, which gives us day and light.
For ニチ, imagine a sunny day perfect for a picnic with a niche cheese: niche -> ニチ, and the sun shines on your day out.