Originally Posted by
TxGreaseMonkey
Remove and fully charge, or replace, the battery; clean the battery terminals and all grounds; and clean the cable connections to the battery, starter motor solenoid, and starter motor. If the problem persists, replace the starter motor solenoid, which is likely located on the fender or firewall. My bet is the starter motor solenoid's contact ring inside is severely pitted, which explains the clicking sound you hear. They run around $20 at Advance Auto Parts.
I'm assuming the "starter relay" you replaced was located in the under hood fuse/relay box.