How much do you want to dig up?
I would start by doing my research. I would look at any papers and documents that your mother will let you look at. I would look at tax records and credit card records and old utility bills and old letters (if any of these things exist). I find that financial records often prod my memory about various events.
Once you’ve done that and read all you can, a million questions will come up because you will have only pieces of the picture. You ask about the pieces you don’t have.
If there are no records, or you are unwilling to do this kind of research, then I would sit down, and write out a history of what I knew about my father. Where he was born, how many siblings, what he did growing up, where he went to school, when he got married, what kind of work he did, what happened at work.
You’ll find many holes in your knowledge. That’s what you ask your living relatives about.
Or you could do both things. You might even end up with enough material for a biography. That might be a nice memorial.
Not that they’ll want to, but if my kids did do that kind of research, they would find it all laid out for them. I kept everything: letters, bills, credit card statements, etc. A lot of it is on computer, too. I just have to remember to give them the unlocking code before I die.