Unless you are a great singer, you will probably have to limit yourself to a song that might seem "overdone". That’s because those particular songs may be ones that people of average or less than average singing ability can manage well enough to have fun. You can bring your own karaoke disc (some KJs with certain computer digital players won’t or can’t play them, but many traditional rigs will). Otherwise, there is usually a song list (usually books of them) of the songs that the karaoke company doing the show has available. There is no point giving you a lesser known, or less popular song, because there is a possiblity that the KJ doesn’t have it.
There is no rule that says that everyone who goes to a karaoke bar HAS to get up and sing. They need people to clap and cheer for the singers too (and it’s considered polite to clap and cheer for anyone who has nerve enough to go up there). Karaoke is for amateur singers. While there are occasionally actual professional singers that might drop in for a lark, and there are plenty of talented regular folks with good voices–trust me–I bet you won’t be the only person that might cause ear bleed! 
The point of karaoke is just to have fun. Sure there are places they have contests, but even contest winners tend to be the most popular singer–not necessarily the best singer–in the house. If a person packs the bar with enough friends and family members they usually will win. People tend to like really familiar songs that they can sing along with as well.
Another thing about karaoke–if you’ve noticed, the KJ can make the singer sound better by adjusting the sound board (adding reverb is one common technique). Did you know that most professional rigs can transpose keys of songs up and down a few steps (not too much or you tend to get distortion)?
Consider rapping. If you can read outloud quickly and keep a reasonable beat, you won’t have to worry about actually singing anything.
Some popular songs from the 1980′s that most companies seem to have:
"I Love Rock and Roll" Joan Jett
"Hit Me With Your Best Shot" Pat Benatar
"Rapture" Blondie (actually not done that often–has a long, weird rap section in it)
"The Tide is High" Blondie ("limited vocal range)
"Bette Davis Eyes" Kim Carnes (lots of sections you can kind of "talk-sing")
"Sweet Dreams Are Made of This" Eurythmics, featuring Annie Lennox (stick to her Eurythmics songs–the others she recorded as a solo are difficult)
"Material Girl" and "Like a Virgin" by Madonna (ham these up as much as you can)
"I Will Survive" Gloria Gaynor (okay it came out in 1978, but close enough–all the women in the bar will sing with you)
‘Walk Like an Egyptian" The Bangles (crazy-azz rhythm, but do it if you know it well, it has a fairly limited singing range)
"We Got the Beat" Go-Go’s
"Mickey" Toni Basil
Hopefully you’ll find a few you can sing. Karaoke song lists tend to be arranged by artist as well as title, so if you like certain artists–say "Cher" you can look up their name and see what songs are available to choose from.
Relax and just let your inner diva run wild that night! I know there are people who take karaoke far too seriously for what it is (usually those singers who think they are better than they actually are).
Jim