Origin and history of e.g.

late 14c., "an instance typical of a class; a model, either good or bad, action or conduct as an object of imitation; an example to be avoided; punishment as a warning," partial re-Latinization of earlier essample, asaumple (mid-13c.), from Old French essemple "sample, model, example, precedent, cautionary tale," from Latin exemplum "a sample, specimen; image, portrait; pattern, model, precedent; a warning example, one that serves as a warning," literally "that which is taken out," from eximere "remove, take out, take away; free, release, deliver, make an exception of," from ex "out" (see ex-) + emere "buy," originally "take," from PIE root *em- "to take, distribute." 

Latin, literally "that is (to say)," from id "that," neuter of is, from PIE pronominal stem *i- (see yon). For est, see is. Usually abbreviated i.e. "to write, or even to say, this in the full instead of in the abbreviated form is now so unusual as to convict one of affectation" [Fowler]. It introduces another way to say something already said, not an example of it (which is e.g.).

gwerə-, Proto-Indo-European root meaning "to favor."

It might form all or part of: agree; bard (n.); congratulate; congratulation; disgrace; grace; gracious; grateful; gratify; gratis; gratitude; gratuitous; gratuity; gratulation; ingrate; ingratiate.

It might also be the source of: Sanskrit grnati "sings, praises, announces;" Avestan gar- "to praise;" Lithuanian giriu, girti "to praise, celebrate;" Old Celtic bardos "poet, singer."