I'm guessing that he used 'summer' because it has the warmest (and therefore best) weather. Given that England has very cold weather, it makes sense that a summer day could have been considered precious there. Having said that, I really don't know for sure. Shakespeare could have meant something else entirely. If it was an expression in Old English, it isn't one in modern English (classical Chinese from around the same time was very different to modern Chinese as well). Or maybe it was just a subtle way of calling someone hot!

Shakespeare's poetry is very open to the reader's interpretation. In my opinion, since summer is the warmest season, he is comparing the woman's beauty to the warmth and joy of summer. The days are longer, the flowers are fully in bloom and many fruit is ripe, etcetera. There are many who believe summer is the best season.