The Latin term religiō, origin of the modern lexeme religion (via Old French/Middle Latin) is of ultimately obscure etymology. It is recorded beginning in the 1st century BC, i.e. in Classical Latin at the beginning of the Roman Empire, notably by Cicero, in the sense of "scrupulous or strict observance of the traditional cultus".
The classical explanation of the word, traced to Cicero himself, derives it from re- (again) + lego in the sense of "choose", "go over again" or "consider carefully". Modern scholars such as Tom Harpur and Joseph Campbell favor the derivation from ligo "bind, connect", probably from a prefixed re-ligare, i.e. re- (again) + ligare or "to reconnect," which was made prominent by St. Augustine, following the interpretation of Lactantius.
The problem with these etymologies, regardless of whether one favours lego or ligo, is that the now-familiar prefix "again" is not attested prior to its occurrence in religio and is itself in need of an etymological explanation.
Within the system of what we would now call "Roman religion (in the modern sense of the word), the term religio originally meant an obligation to the gods, something expected by them from human beings or a matter of particular care or concern as related to the gods, "reverence for God or the gods, careful pondering of divine things, piety".
In this sense, religio might be translated better as "religious scruple" than with the English word "religion". One definition of religio offered by Cicero is cultus deorum, "the proper performance of rites in veneration of the gods."