The word midsummer comes to us from Old English, and it has a Dutch cognate midzomer, and Scandinavian cognates (e.g. midsommar in Swedish), so it may even come from an older Germanic language. Both the old Anglo-Saxon calendar and the old Icelandic calendar had two seasons, summer and winter. For these calendars, "Midsummer's Day" would have fallen near the middle of summer (probably not the exact middle ... summer started in mid-April in the old Icelandic calendar, and on a full moon in the old Anglo-Saxon calendar).
The Anglo-Saxon calendar also explains why summer and winter are words which have roots in Proto-Germanic, while fall and spring were not used for the seasons until Middle English, and autumn is originally a Latin word.