I have done this a few times and decided to document it the last time for guys that have not done it.
[Copyright release from Mike Meier for this BBC informational post only. 2/2/15]
<Moderator Edit: Some of this post has been edited, to ensure appropriate images are in use, and provide additional credit and information. MANY tanks to Mike for his hard work on this post!>
According to the Mercury Service Manual:
"Cooling water strainer screen cleaning.
Remove lower cowlings. 4 bolts. Remove the cowling seal.
*The lower rear bolt is accessed by removing the rubber plug on the port side cowl. If it’s cold, a heat gun/hair dryer will help to soften it up."
Note the RUBBER ACCESS PLUG inserted in the LOWER COWLING (port side) at Location marked "C" in the below images:
A: Inside FRONT of cowling, port side, at very FRONT.
B: Outside of cowling, Port side, UNDERNEATH near front
C: Remove rubber ACCESS plug, screw is INSIDE cowling, down low (access through plug opening)
D: Inside rear of cowling, port side, up high.
The port side cowling section will remain intact due to the incoming cables but can be moved away from the engine to increase your work space. The starboard side cowling can be removed once you disconnect the trim switch cable.
Locate the cooling water hose. Remove the retainer & hose from the strainer screen insert.
Remove the strainer using a deep well socket. Clean the inside of the threaded hole and wipe dry. Clean the screen.
It is sometimes helpful to connect a pin-style earmuff attachment, cover the LOW water inlets, and WHILE WATER IS SUPPLIED, start the engine and allow it to idle (strainer REMOVED). DO NOT RUN THE ENGINE FOR MORE THAN 30 seconds with strainer removed.
This helps to flush away remaining debris trapped in the opening (or the threads of the adapter plate), as a large volume of water will escape through this opening during the above (30 seconds total/maximum flush procedure).
Apply pipe thread dope.
Install strainer. Attach hose using Part# 816311T "Radius Head Tie Wrap" (if using a tie-wrap gun, set it to 50%).
Once strainer has been reinstalled, and appropriate clamp installed on the hose, REINSTALL the lower engine cowling (installation is the reverse of removal).
Test run the engine on an appropriate flush attachment (don't forget to cover the low water inlets of the gearcase, as appropriate), and check for leaks.
Also verify appropriate idle water pressure, and good TELLTALE stream are present.
Reinstall upper engine cowling once proper function has been verified.