They hadn't given him a formal name, because he was only 4 weeks old and they hadn't decided WHAT they wanted to name him. As I recall in the TV series, Laura Ingalls Wilder and her husband called him "Baby Wilder".. That wasn't unusual in those days. There were no "birth certificates" to be filled out when a baby was born.. The baby's name was written in the family Bible when the parents decided upon a name. .
In the TV show, their baby died suddenly in his sleep from what is now known as SIDS (Sudden Infant Death Syndrome) - for which there is often no explanation, even today. The Wilders' baby died in his sleep, and his parents found him dead in his crib the next morning. When they put him in his crib the night before, he appeared to be a normal, healthy 4-week-old baby.