Al,
Your lifters are probably what's causing the clacking, chattering, clicking, tapping noises you hear. If you only notice the tapping noises from a cold start or before the engine gets to a normal operating temperature, you probably don't have to worry about replacing them. But if you always notice the noise whether the engine is warm or cold, replacing the lifters will probably cure that noise.

If you valves are what's causing the noise, here is what Mitsubishi mechanics would suggest as probable cause and remedy:

Probable Cause Remedy
------------------------- ---------------------
1. Incorrect lash adjuster Bleed air or replace lash
adjuster

2. Thin/diluted engine oil Change engine oil
(low oil pressure)

3. Valve stem or valve guide Replace valve and/or
worn or damged guide

These are taken right from the Mitsubishi Eclipse Factory Service Manual.

Any valve or lifter work to be done is going to take some time. If your belts are in good shape and you don't plan on replacing them, be extra careful when removing them. Get yourself a Haynes or Chiltons manual if you plan on doing the work yourself.

Good luck!
Anthony