Official Naming of the Characters (Hanzi) in Mandarin Chinese
Generally, there is no name for a single character in Chinese. Or you can say the name is the character itself, but then there are too many duplicates when you only have the reading.

To address a single character, there are often two ways, one is to describe how it is written, one is to put it into a word.

Take "博" as an example. You can say that it is the "bo2" with a "十" on the left, which is describing how the character is written. Or you can say it is the "bo2" in "博士"("博士的博"), which is putting it into a word.

For some characters, there are generally accepted ways to describe them. For example, "李" is often referred to as "木子李", and "张" is often referred to as "弓长张".