Which board game is the first to be ever invented, when was it invented and where?
This academic paper makes a strong claim for Three Men's Morris (later to become Nine Men's Morris; the game is generalised to 'merels' in the paper); it may be that some Stone Age or Bronze Age petroglyphs were boards scratched on boulders near houses. R.C Bell, in Board Games from Many Civilizations, claimed that a Three Men's Morris board was scratched into an Egyptian temple about 1400 BC, but it is fair to say this is not universally accepted.
Of course, the definition of a board game is somewhat vague. One petroglyph was apparently used in adjudicating 'court cases'; each side threw a stone over their shoulder and tried to hit the design, with the closer to the centre being approved by the gods, so deemed to be telling the truth. This is no longer used in jurisprudence (to the best of my knowledge), but could be considered a game...
The oldest games that could be considered "board" games are almost certainly of the Mancala family.
Mancala pieces have been found throughout Africa (including Egypt) and The Middle East to dating all the way back past 6000 BCE, with the earliest potential Mancala board (Fig A) coming from ~7-6500 BCE at ʿAin Ghazal, but it's possible that two rows of divots were carved into a stone for an unrelated reason.
Note: Fig B. below is from Qal’at al-Bahrain, and dates from ~3-3500 BCE.
And therein lies the main issue with dating when and where Mancala style games first developed, as a rudimentary board can be made by simply digging a number of holes in the ground, and collecting random items from the surrounding area as playing pieces (shells are a popular option).
The simplicity of board creation also leads to a secondary issue with dating Mancala's origins, as even when the surrounding materials can be dated, a board can easily have been carved much more recently by later occupants; an example of this is the board found on the roof of the temple of Seti I (1290–1279 BCE) in Egypt, which was most likely carved by priests using the temple roof for astronomy sometime after construction.
Not to forget Backgammon which is supposed to be one of the oldest games in existence. Around 5000 years ago it appeared in the Mesopotamian area. Until today experts are not sure if that was really a backgammon game or a distant relative or even a completely different game. The fact is that there was a board game in ancient Persia that was at least similar to our backgammon. From there, the game probably came to India and Egypt - in exchange for another time-honored game: chess. Archaeologists also unearthed a backgammon-like game board in the lost Mesopotamian metropolis Ur.