Should it be
The weather was fine during (or at) the time of the accident.
Accidents are normally things that happen "at" a point in time. Use "at" to indicate a single point. If something takes a longer time, you can use during. For example.
The weather for fine for during the majority of the flight. But at the time of the accident a sudden storm had developed and visibility was poor.
The correct choice of word here would be "at", and not "during".
The weather was fine at the time of the accident.
"During" is used for events that happen over a duration of time (hence "during"). "At", on the other hand, is used for events that happen at a particular time. Going by this, we can also use "during" for an accident, provided we are talking about something that happened over the course of that time. For example:
The car flipped over 7 times during the accident.
In this context, we would use "during" instead of "at".
during and at can be used in different circumstances: weather during the accident implies the elapsed time between the start and end of the accident, while at will imply the weather condition during the instant of the accident, like the moment when the cars hit.
However, you can use both instances almost interchangeably in commonplace speak, as people will know what you're talking about.