Check your Serpentine Belt First
 

I had a squeaking/rattling that sounded like a bad tensioner bearing or water pump on my 05 Toyota Matrix. Water pump is only 1 yr old and was replaced trying to cure this squeaking which occurred w/ cold engine and went away warmed.

I also removed tensioner recently and repacked tensioner bearing (it was dry), which also failed to cure problem.

Yesterday, I was replacing spark plugs (120K maintenance) and noticed the squeaking/rattling again when I restarted after installing new plugs.

I'm thinking I've got either a bad water pump again, or bad alternator as the noise appeared to be clearly from front side of engine.

Then I noticed the serpentine belt had lots of splits and gaps where material had disappeared. So I decided to replace the serpentine belt.

You guessed it! The squeaking/rattling was cured. Apparently the belt had stretched and worn enough to reduce tension allowing a cold engine to squeak/rattle. Probably the water pump drive was slipping on worn belt.

Lesson is to take a hard look at serpentine belt, and it it's old, it may be cause of apparent tensioner/water pump noise.

Regards