A Brief History of Swearing an Oath on Objects
The practice of swearing an oath while laying hand on an inanimate object can be traced back centuries. Most commonly, we think of the Bible in such cases, but that is only because of a tradition that the English court started in the 12th Century. Long before that time, it was common for a knight to swear by their honor and/or sword, or for a regular person to swear by whatever deity they held dear. The object being sworn on was typically determined by whichever party asked for/demanded/initiated the swearing part. Whichever side wanted the swearing would choose an object sacred to THEM, but not necessarily both parties.
The general idea behind this concept was that if a person broke their oath, they would be punished by whatever superstitious conceit was included in the ceremony. If a person swore by their honor, for instance, then later broke the oath, it was expected that the other party would make the dishonor known. A businessman, for example, would find his reputation as an oath-breaker preceding him to other towns or regions. A knight who swore by his sword would be risking the integrity & strength of his blade if he broke the oath - something the notoriously superstitious warriors of early history would not want.
Similarly, a 13th-Century court manuscript from England states the idea behind using the Bible to swear in witnesses:
By placing a hand on the book and then kissing it, the oath-taker is
acknowledging that, should he lie under oath, neither the words in the
Bible nor his good deeds nor even his prayers will bring him any earthly or
But why the Stone of Erech?
To answer your question more concisely, we don't know exactly WHY Isildur chose the black stone for the swearing ceremony. It was obviously an object held sacred to the Numenorean people, and may have even had magical properties that aided Isildur in his curse.
Also, it's important to note that Isildur did NOT bring the stone specifically for the oath. The stone was originally a souvenir of Numenor that they had brought with them and planted in the ground as a memorial. Years later when the Kings met Isildur at the Hill of Erech, the stone was the only standing stone large enough to meet and/or swear on. It may have been a completely random choice by Isildur based on where they were, or the meeting may have deliberately taken place at the Stone of Erech for that very purpose.
As for the stone itself, we don't know much about it, simply that it was a large spherical stone, buried halfway in the ground and polished to a high sheen. The only real description we have of it can be found in Return of the King during "The Passing of the Grey Company":
...upon the top stood a black stone, round as a great globe, the
height of a man, though its half was buried in the ground.
Lord of the Rings Online depicts the stone as very large, three times the height of a Man:
This depiction seems overly large compared to Tolkien's description. Most fans have interpreted the text to mean that the exposed portion of the stone is roughly 3 meters tall (the height of a man), while others believe that comparison refers to the entire, unburied stone (meaning that the exposed portion would be only waist-high). Either way, the Stone would obviously function as a permanent landmark - something large enough to go on maps and assumed to be around for centuries if not millennia.