It implies that the name is familiar, but the speaker doesn't have a strong memory of the person or their connection to them.
Here are some examples:
* "I met someone at the party last night, but I can't remember their name. I think their name was Emily, but it just rang a bell."
* "The name of the author on that book rang a bell, but I couldn't remember if I'd actually read any of their books."
The idiom is based on the sound a bell makes when it is rung, which is a sound that is familiar and easily recognizable. In the same way, a familiar name can "ring a bell" in someone's mind, even if they don't have all the details.