iMac g3 white power light flash

You can always try replacing the internal 3.6-volt ½AA battery on the motherboard, but the symptom that your dog detected with his nose may indicate a more serious problem. Unfortunately, your convection-cooled (fanless) slot-loading iMac is exhibiting the same symptom that mine did, after having worked perfectly the last time it was powered ON a year prior. The most common failed component in those iMacs is the Power-Analog-Video (PAV) board. It's the heart of the power system for the computer, but eventually most of them fail, depending on the hours of use the computer has had and where it has been stored. New boards aren't available and used replacement parts have unpredictable remaining service life, which can make the time and $ invested in buying/installing one a waste of resources. The inherent danger of working in that area of the chassis involves the shock hazard from the stored high voltage, which can be lethal for some time after the computer has been in storage. Proper discharge of the CRT is required, before the PAV board can be replaced. Given the age of the computer, it may be time to retire your iMac, as I decided to do with mine.

Just a couple of comments in addition to the information already provided by Jeff.

Someone well experienced in doing high-voltage equipment repairs (for example, an old-school TV technician) might be able to locate the offending electronic component. The iMac's power/analogue/video area is not easily accessible (for a reason), and users should never carry out any work here.

Electrolytic capacitors do often create problems in older computers. Some paper capacitors (typically, used in power line filters) have also been known to break.

Remember that many capacitors, working at high/line voltage levels, and even at lower voltages, can carry very dangerous charges, so be careful.

When you change the 3.6 V battery, you may at least want to inspect the logic board (using a magnifying glass). Look for cracked/discoloured/leaking components.