I found this at allexperts.com. It, mostly, corresponds with what I was taught in art school:
Lowercase letters, also called ‘small’, ‘minuscule letters’ in contrast to the majuscule, or uppercase, capital letters, which were the earliest form of writing in ancient alphabets, date back to the 3rd. century BC in Greece where in fact the lowercase letters were used to write faster on papyrus and parchment.
The different shapes of the lowercase letters developed gradually through transformations of the capital letters by the elimination of a part of the letter (see ‘b’-beta in Greek- from B) or by lengthening a part of it
( see ‘d’ –delta in Greek- from D).
As for Latin alphabet which derives just from Greek alphabet which in turn derived from the Phoenician one dating back to ca. 1200 BC, it is still today the standard script of the languages of most of Europe and those areas settled by Europeans, with a few exceptions like the Greek alphabet used in Greece and the Cyrillic alphabet (Russia, e.g.).
In ancient Roman times too there were only capital letters, while it seems that lowercase letters have been introduced in the 1st.century AD.
This change from capital letters to small letters which was influenced by the nature of the material of writing (papyrus, parchment and later paper) continued in the Middle Ages, of course.
To conclude, the origin of lower case letters does not pertain to Latin and the Middle Ages, but dates back to the ancient Greek in the 3rd. century BC, though the use of lower case letters was implemented in the Middle Ages.