Does Mom have old photographs? If you were to go through them with her, that might trigger memories. When I first started being interested in genealogy, I asked my Grandma what her parents and grandparents names were. She gave me her parents, but her grandparents was, "It was so long ago, I can't remember, why do you care anyway." A couple years past, and I told her some stuff about my Grandfather's side of the family, and then she asked if I could research her side! She gave me the names of her maternal grandparents, aunts and uncles, cousins. She didn't know anything for her paternal grandparents because her father died when she was 6 and his family didn't like her mother so she never saw them.
But maybe if you ask a second time, and maybe come at it from a different angle. Ask about her childhood, what she did, and then if she mentions family members you can ask for names. And you could, always, have her request her own birth certificate, if she is willing. Go the website of the state of her birth; in many cases you can order online. Once you get back to people that were alive in 1930 (soon 1940) it gets easier, because you can search the Census Records. Are there any aunts/uncles still alive that you have contact with?
I know my Grandma did not have a good childhood, so I think talking about that stuff brought up bad memories she didn't want to remember. However, she and her sister are the only ones left of their generation, and now they talk about "old times" so now things are in a different context. I had set my genealogy stuff aside for awhile, but when I was in FL, she called and wanted to know if I knew her father's birthday. I didn't, but as soon as I got home I sent off requests for her Dad's death certificates, and also her maternal grandparent's death certificates. I just hope they can find her Dad's. There is an online death index for NYC, which is where I found the certificate numbers for the grandparents. But no record of her Dad. I hope it's just an improperly transcribed name (I haven't seen a correct one in any of the Census' so certainly possible.). I have the death date, so it should be "findable."