It is important to note here that "difficulty" can be used to describe a condition, or a situation. The expression "breathing difficulties" could be used as if it were a person's medical diagnosis, but it could also be used to describe a transient situation, or something that they experienced perhaps only occasionally, or even once. The word "complication" is similarly used as both a circumstance and a medical condition.
In this context then:
Her mother observed him have breathing difficulties shortly before death implies that her mother witnessed an instance, or multiple instances of breathing difficulties.
Her mother observed him to have breathing difficulties shortly before death implies that he had breathing difficulties in the sense of possessing them. This would indicate that perhaps it was an acquired or permanent condition of his rather than just something that happened on one occasion. In this context "observed" could also mean that this was noted over a period of time rather than a single incident that was witnessed. It would not necessarily follow that the breathing difficulties were actually occurring at the time of his death, just that he had that condition.
Her mother observed him having breathing difficulties shortly before death implies, like the first example, that instances of breathing difficulties were witnessed. The only difference here is that having is the present participle so along with "shortly before death" this example might better imply that these difficulties led up to the moment of death.