Aren
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Aren was the name of the ring that Brom bore. It had the yawë inscribed on it, signifying the wearer was an Elf-friend. It became one of Eragon's possessions after Brom's death.
According to Queen Islanzadí, Eragon should not have worn it, as it was not gifted to him directly. However, due to Eragon saving Arya and all that he had done, the Queen bestowed the ring upon him and named him an Elf-friend.
In Brisingr it was revealed that, during Brom's self-imposed exile in Carvahall, he squirreled away every bit of energy he could spare into the sapphire jewel. It also stated that after Eragon washed away all of his aches and pains, he found that he had taken very little power from the ring.
[edit] Speculation
Aren's sapphire was initially speculated to be Saphira I's Eldunarí (Heart-of-hearts). However, the author disproved this in an interview, clarifying that Aren had been made by the elves, and that it was expertly crafted to hold extreme amounts of energy.
There was also a time lapse problem. Brom gave Jeod Aren in Teirm, to prove that he was contacted. However, in the memory Brom left Eragon, he was seen with Aren on. It was established that this memory took place after the scene with the Urgals, which was AFTER Teirm.
