Grammatically, "a perfect start of the day" makes sense. And without the word "perfect", it would usually be the best choice, as in "at the start of the day".
But there is an established idiom for phrases like "a good start to" when describing some period of time or event, where the adjective good can be replaced with many other adjectives describing value quality: great, perfect, excellent, bad, poor, disappointing, etc.
Since probably most English speakers are used to this idiom, it will be better to say "a perfect start to the day".
(Interestingly, it looks a century or more ago, "a good start of" was more common than "a good start to", but "a good start to" has been steadily becoming more frequent: see the Google ngram comparison.)
I will put this sandwich aside and eat it later in the day.
The implication is that I will eat this sandwich later in one sitting.
I will put this sandwich aside and eat it later during the day.
The implication is that I will take a bite or two from this sandwich starting later on and maybe a few more bites again even later.
Later in the day (month, year, season, whatever) refers to a time that falls within the span of the day, whereas during the day refers to a time-span that falls with that span, or to several times falling within that span.
Best Answer
Grammatically, "a perfect start of the day" makes sense. And without the word "perfect", it would usually be the best choice, as in "at the start of the day".
But there is an established idiom for phrases like "a good start to" when describing some period of time or event, where the adjective good can be replaced with many other adjectives describing value quality: great, perfect, excellent, bad, poor, disappointing, etc.
Since probably most English speakers are used to this idiom, it will be better to say "a perfect start to the day".
(Interestingly, it looks a century or more ago, "a good start of" was more common than "a good start to", but "a good start to" has been steadily becoming more frequent: see the Google ngram comparison.)