How do you know which word to follow it up with? "The kind of _____ where/when/on which/during which/that" etc.
I would always say, "It's the kind of a morning when... ." Morning is a time of day, and "when" is the conjunction used to refer to time. "Where" is never correct in referring to time. "Where" is used to refer to place, for instance, "It's the kind of a restaurant where... ."
"The morning on which he was born" and "The morning that he was born" are both correct, and have the same meaning. However, "The morning during which he was born" suggests that the birth was a long and difficult one. "During" has a more progressive nuance than "when." Things take longer during long vacations, but things go well when time is short.