What does the eyeball button in the options tab do

Prestiging in Jetpack Joyride means you have completed various given missions and earned enough stars to level up past level 15. The levels 1-15 require earning progressively more stars to move to the next level. The missions to earn stars are things like:

• Fly 300M in the Profit Bird in a Single Run
• Rub your head against the roof for 1000 meters
• etc.

If you level up past level 15, you will be given the option to Prestige. Once you do so, you get a shiny badge on your profile, and then you should be able to access the new missions in the update.

After you Prestige, Level 16 is like starting back out at Level 1, when you hit level 30, you can Prestige again, rinse and repeat. I do not know if there is a cap, but I'm at level 57 currently.

According to this question on Android.stackexchange.com it looks like you cannot force a re-installation through the market, but what you can do is try to force the Market app itself to re-synchronise your installed apps.

Go to Settings -> Applications -> Manage Applications -> All -> Market and select Force Close and Clear Data, then restart your phone and open the Market app. This may work, but is not guaranteed.