This expression is used to pick one thing from a particular "genre" of things and suggest it lightly. The speaker doesn't mean to be specific about his/her suggestion.
お茶でも飲みませんか。
Why don't we have some tea or something?
明日でも小林さんの家に電話をかけてみましょう。
Let's call Mr./Ms. Kobayashi at his/her house, maybe tomorrow.
If N is followed by other particles such as [place] へ and [person] に, でも is placed after these particles.
あしたは新宿へでも行きましょうか。
Shall we go to Shinjuku or somewhere tomorrow?
この本は弟にでもやろうかと思っています。
I think I'll give this book to my younger brother or somebody.