Due to many hidden easter eggs throughout the show, the fact that the mother has never been shown in 2030, and a scene in an episode in season 9, I believe that the mother has an illness and will be dead in 2030.
The children were disinterested when the father started with the story, They were like "Are we being punished for something". I believe that 2 children living without their mother MUST be interested in the story.
I also believe that the reason why he wanted those 45 minutes so much in 'The Time-Travelers' is because he is more further apart from others and lonelier than ever.. He finds himself alone with no one to love or care for.
There isn't any other reason to believe in this fake theory which is being circulated.
In 2030, we never see the mother. While this story is probably shorter than 9 years, in the while it would take to tell it you think that the mother would be heard or seen. In the episode "vesuvius", during a conversation with the mother about Robin's mother, the actual mother says, "what mother would miss her own daughters wedding." This causes Ted to start crying. This could have been due to the death of Ted's mother, but Ted is not a girl (as far as I know). It might instead refer to a terminal/chronic illness the mother has, and they both know that she will be dead before the daughter has her wedding.
Also, a recent death of the mother might have explained Ted's urge to tell this story to his children.
Finally, he always refers to the mother as "your mother", and not "my wife." This could be just his preference, or it could be more proof the mother is dead in 2030.