There are a number of hadith in Sunni and Shi'i books, especially Shi'i books, on the merits of wearing rings, especially an 'aqiq (carnelian) ring on the right hand.
It is narrated that the angel Jibra'il descended to the Prophet (S) from Paradise and said, "O Muhammad, wear an 'aqiq ring, and command your ummah to do the same."
It is also narrated that the angel Jibra'il descended to the Prophet (S) and said, "Your Lord conveys his salaam," and then told the Prophet (S) to wear an 'aqiq ring and command his cousin [i.e. Imam 'Ali (A)] to do the same.
A narration says that this is because 'aqiq was the first stone to bear witness to Allah's lordship, the Prophet's prophethoold, and Imam 'Ali's successorship.
While many Muslims do wear religious rings, over time, it became particularly associated with the practice of the Shi'a and the Imams, and so one can say it became part of the tradition. Perhaps this is why wearing a ring is described in a narration as one of the signs of a believer.
The Prophet (S) also was said to have worn a signet ring, i.e. a ring with a seal which is used as a a stamp on wax to sign a letter, as was a common custom in the past. So, wearing a ring (albeit this kind of ring without a stone) can also be seen as a Prophetic practice from this angle.
Additionally, there are a number of narrations (correctly or incorrectly, God knows best) attributed to the Imams (A) describing the merits of wearing specific stones. For instance:
* 'Aqiq (carnelian): strengthening faith, protection, protection from poverty, dispelling melancholy
* Turquoise: protection from poverty, acceptance of du'a
* Hadid sini (hematite): overcoming fear
* Ruby, emerald, and sapphire: overcoming poverty
* Durr-e-najaf (a type of quartz) is also said to have many blessings and to have been made an affordable stone for the followers of Ahl al-Bayt (A).
Anyway, it is good to wear a ring with the intent of following the tradition of the Prophet (S) and Imams (A); or out of hope that it may offer these benefits, spiritual or material; or even if one simply likes it. (After all, Allah is beautiful and loves beauty.) However, it is better not to get too caught up in these things and put faith in the material over the immaterial; that is, faith should first be in Allah, and physical things such as rings are only secondary factors. It is also fine if one prefers not to wear a ring.