Hi guys,
I have a 2001 Chevy Tahoe, and for the past couple of years, I have been having an incredibly frustrating issue with my car. When I go to turn the key, it sounds as if it is about to turn on (ie the engine is about to turn over and it sounds normal) then right before it catches, it will start clicking, the noise sounds like the battery is dead. When it first started happening, I called AAA, however by the time they reached me, the car had started again. Now when it occurs, I wait 10-15 minutes and it will start- every time. I have had it in to 4-5 different car repair shops, however no one can seem to figure out what it is. I have had the battery replaced, and several other things altered, etc, however nothing seems to fix the problem. I am not sure where to go from here, I was informed to check the fuel pump and/or filter yesterday- it happened twice on me yesterday as well. Any suggestions?!? I’m helpless! 
Make certain ALL the battery cable connections are clean and WRENCH tight.
Do a bench test on the starter to check the solenoid and bendix.
There MAY be an ignition system fault where something heats up during the initial attempt to start but gets too weak, then after a short cool down period, temporarily rights itself.(guessing)
Check the underhood fuse and relay box for a start relay. If it has the same part number as another relay, just swap relays to see if the problem goes away. If it does, replace the relay.
Make sure the large cable connected at the starter solenoid is tight, the nut retaining this cable is frequently loose, same thing goes for the large ground from the battery to the engine block. I hope the mechanics you have had the car to know how to spot damaged battery cables (from a leaking battery)