Yes, it is normal if there is a baby or a young child in need of diaper - wearing in the family. Moms often take on the role of diaper - changing and putting diapers on their kids.
Perhaps they were preparing for a long journey. Diapers are convenient during travels, especially for small children. The mom would have chosen a suitable diaper, probably one that is absorbent and comfortable. She would then put it on the child, making sure all the edges are secured so that there are no leaks.
One interpretation could be that the child is sick or has had an accident and the mom has to put a diaper on them for practical reasons. For example, if the child has a stomach bug and can't control their bladder or bowels well.
There was a time at a baby - sitting gig where the baby was so active that the diaper got loosened. I had to quickly put it back on. I first made sure the baby was in a comfortable position, then adjusted the diaper to fit snugly. It was a bit of a challenge as the baby was constantly moving, but I managed in the end.
There are no appropriate ABDL stories to recommend as this concept is not suitable for general promotion. We should encourage healthy and positive family relationships in a proper way, such as stories about a mother teaching her child good values or sharing happy family moments.
I'm not sure which specific 'friends diaper story' you are referring to. There could be many stories among friends related to diapers in different contexts, like friends taking care of a baby together and dealing with diaper changes, or perhaps some funny incidents during a baby - sitting session among friends.
A possible plot could be that two friends volunteer to babysit for another friend. When they start, they realize they have no idea how to change a diaper properly. They fumble around, make a mess, but in the end figure it out and have a good laugh about it.
Once upon a time, there was a little boy named Tom. His mother put him in a diaper as he was still a baby. Next door lived a girl named Lily. She loved wearing her pretty dress. One day, their families had a picnic together. Tom was lying on a blanket in his diaper, looking at the sky with his big innocent eyes. Lily, in her dress, danced around the picnic area, making everyone laugh. It was a simple and sweet day for both of them.