M.L.
It's not unusual for one-year-olds to want to stick to Mama like velcro. As I've written on some other posts, I have a marvelous picture of my youngest granddaughter sitting in a gorgeous field of Texas bluebonnets, crying as if there were no tomorrow, because her Mama SET HER DOWN! Horrors!
It WILL pass. The girl may have to fuss a little. Your friend may have to use some diversionary tactics to persuade Miss Velcro to sit by her for a bit while she does something else. She needs to keep her sense of humor and to try not to get frustrated. "There! That didn't take long, did it? And you did a fine job of playing with that toy while I was helping your brother/put the laundry in the washer/used the bathroom." When my children went through that stage, I found that playing peekaboo games helped a bit, too.
When Miss V. finally grows a little more, and has it in her head and heart that if Mama disappears she will be back, then she'll will be more comfortable. Pretty soon your friend will have to go looking for HER.