〜でも(例示)
… or something (gently offering an example)machine translation
FormationNoun + でも
Noun + でも here is used to OFFER AN EXAMPLE in a gentle way: "tea or something", "a movie or something". お茶でも飲みませんか literally means "won't you drink tea or something?" — the speaker puts お茶 forward as an example rather than limiting the choice to exactly that item.
Thanks to this "leaving the options open" nuance, 〜でも makes invitations and suggestions sound much softer and more polite: compared with お茶を飲みませんか (inviting tea specifically), お茶でも飲みませんか makes the listener feel more at ease. This is a very natural way to extend invitations in everyday Japanese.
Don't confuse it with the sentence-initial でも "but" (learned at N5) or with 疑問詞+でも (何でも: anything at all): in this pattern でも comes after an example noun and takes the place of the particle を・が.
Examples
お茶でも飲みませんか。
Would you like to have some tea or something?
暇なら、映画でも見に行きましょう。
If you're free, let's go see a movie or something.
週末は本でも読んで、ゆっくり休みたいです。
This weekend I want to read a book or something and take it easy.