Sometimes, the simplest place to start when beginning to craft your character's backstory is right at the beginning of your character's life. Unless your character was somehow created as a full grown adult - such as a robotic lifeform (android, robot, clone, created as a full grown adult by a diety, etc.) - they more than like had parents who gave birth to them. While science (both reality and fantasy) have not yet been able to adequately define how two individuals of the same biological sex (not including gender identity or sexual identity) would be able to conceive a child together, the parents of your character will typically include a mother and a father; depending on biology.
With the exception of characters who were orphaned or given up/away at birth, most characters will probably have spent a number of years being raised by one or more parents (e.g., biological or adoptive). And it is with those parents that your character will first begin to discover who they are as individuals.
To start out with, create a family for you character. Families don't always have to be cuddly and loving. Sometime they can be quite damaging to your character. Regardless of the familial environment, this is where a character will receive their birth names and family names. Some characters, however, may choose, for various reasons, to change their name later in life. But for the purposed of this prompt, focus on the name your character was given at birth in order to discover who their parents are or were.
Using tools like Fantasy Name Generator, or by coming up with the names on your own, explore your character's parents' name. After giving them names, you can start to explore the possibilities of any special meanings behind those names. Are the names of your character's parents or your character handed down from past generations (e.g., named after a grandparent or some other relative), or do the names have any special or particular meaning within your character's community or culture?
If by chance there is some kind of special meaning or purpose found in those names, make sure to communicate that to your DM/GM as they may be able to help craft a unique storyline within your game that centers around your family and those "special" names.