From the Super Mario Wiki
“Egad! My baby!! Have you seen my baby??? She´s the most precious baby in the whole world. They say she has my beak...I just can´t remember where I left her. Let´s see...I stopped for herring and ice cubes, then I...oohh! I just don´t know!”
Tuxie's mother is a penguin who appears in Super Mario 64 and in Super Mario 64 DS in Cool, Cool Mountain. She has lost her daughter, Tuxie, so Mario goes to look for her. Once Tuxie is rescued, Tuxie's Mother will reward Mario with a Power Star.
If Tuxie is taken back after being delivered to Tuxie's mother, she will follow Mario with a cross expression until he drops Tuxie again. However, she does not seem to react in any way if Mario gets far enough away or drops Tuxie off the edge.