Yuk, you're not gonna like the solution to this but you have one of two things going on. That clicking noise you're hearing is the laser eye hitting the disk cuz it's havin problems reading it. There are a couple of reasons that would cause this. 1.) The laser lens is out of alignment. 2.) The lens itself is either damaged or the potentiometer pots on the back of the lens itself (if the ps2 dvdrom drive is taken apart and the eye is removed and flipped upside down there are two screws on the back of the eye, one controls laser voltage for dvds one controls laser voltage for CD) needs adjusting.
If you're lucky it's 1, that's a pretty easy but time consuming fix. You take the cover off the PS2. You power on the PS2 open the dvd-rom via the button on your console then manually power off the PS2 via the switch on the back, thus leaving the dvdrom tray open. You then take the screws off of the top of the dvdrom drive and remove that plate the screws are holding down. You'll see a large COG gear that should appear white in most models. Whether it is white or black there is a allen wrench hole in the center of it, you'll notice. The size of the cog is question is about the size of a 50cent piece(in most models, some it's about the size of a quarter). It's quite large and hard to miss. Get yourself a permanent marker and find some way of marking it's current position so if all else fails you can re-align it back to it's current position. Next get an allen wrench that fits the whole in the center. Here's the time consuming part, Turn that cog (start clockwise) One click (meaning turn it till you feel one click) put a game in, place the plastic lid back on but don't put the screws in and put something on top of the plastic dvd cover to hold if firmly in place and power on the ps2. If the game doesn't load, rinse and repeat this method from the beginning untill you have gone through every single gear notch on the cog. If you're not savvy with written instructions and need pictures to guid you in taredown and identification here's a guide with pics. http://arstechnica.com/articles/paedia/hardware/ps2.ars/1
Now if you've gone through every single gear on that cog and still can't resolve the issue the other steps would include adjusting the voltage on the back of the PS2 lens. I have created a solution for someone else with a similar problem that can be found here http://www.fixya.com/support/t1122813-ps2_modded_trouble_reading_dvdrs
You first use that guide to identify your PS2 version then you go to the link provided within that guide and follow the directions to set your voltage correctly.
If neither of these solutions solved your issue then your PS2 laser lens is damaged totally and needs to be replaced. You can either replace the lens itself (about 20-40bucks) or replace the entire DVD-ROM unit (not needed, but some people do this if they don't feel like dealing with alignment)
Hope that was of some assistance,