Examine the terminals to see if there is any corrosion. If the terminals look fuzzy, clean them with a cotton swab dipped in ammonia. Put in a new battery. If the display is still blank, it is time to give up unless you are handy with a soldering iron. When the battery is replaced, power is restored to the computer.
Some units are a bit fussy about how cleanly power is restored.
Ideally, the battery should make instantaneous, solid, contact with it's terminals, but in some cases, this is not so easy to do. The unit may show all 8's, or random fragments of characters, or may come up blank. This is a particularly common problem with the Cateye Solar , which has very tight spring contacts and uses two small batteries that are a bit hard to install. Many of them have been discarded as defective, when all they need is to have the batteries re-installed correctly. Genuine Cateye batteries, which are no longer available, came with a special little piece of thin plastic that can be inserted between the two batteries and then yanked out to establish contact cleanly.