Praise be to Allah, and may peace and blessings be upon the Messenger of Allah. To proceed:
There is nothing wrong with using the word “coincidence” if what is meant is that a person met someone without a prior appointment, agreement, or intention to meet, and he does not mean that the event occurred outside the decree of Allah, the Most High.
The term “coincidence” has also been used in hadiths, such as the statement of Anas, may Allah be pleased with him: “I happened to meet him, and he had a branding iron with him,” and the statement of Laqit ibn Sabirah, may Allah be pleased with him: “We did not find him at his home, and we happened to meet Aisha.”
Accordingly, a person’s saying, “I met so-and-so by coincidence,” is not considered shirk or forbidden, as long as he means that he met him without prior arrangement or appointment.
However, a distinction should be made between two matters:
With regard to a person: He may meet someone without knowing about the meeting beforehand or expecting it, so it is permissible for him to say, “I met him by coincidence.”
With regard to Allah, the Most High: Nothing occurs by coincidence in the sense that it is outside His knowledge and decree. Allah, the Most High, knows everything, and everything that occurs does so by His knowledge, decree, and will.
Therefore, the word is permissible in this customary sense, and there is no need to replace it with another expression. One may also say: “I met him without an appointment,” or “I met him without any prior arrangement.”
And Allah knows best.