I had the same problem with my PS2. At first, it started having problems loading up anything DVD. After a couple of tries, it would finally load the game up, and it would seem to work fine. However once the unit was completely turned off, and a couple of hours passes, it became a hassle to load up dvd games once again.
I too bought a cd/dvd lens cleaner, and it seemed to fix the problem for a short period of time, but did not ultimately fix the problem. Knowing a little bit about CD players and lens cleaners, I can assure you that lens cleaners do not do the job. Basically they\'re supposed to dust away the dust that ends up on the lens, however often times, the lens does not get cleaned much at all. If the lens gets too dirty, then the cd player does not recognize the lens cleaner as a viable cd, and will not load at all.
So basically I aborted the lens cleaner and decided simply to take the ps2 apart and dust the lens myself. However I do not suggest that everybody does this because I have scratched many cd players before, and in effect, deemed them useless afterwards.
Methods that I have tried before which seem to work well:
Take the ps2 apart. I forgot exactly how many screws there were on the bottom(I\'m currently at work) but take the outer casing fof, and you\'ll find that the cd/dvd drive unit is a separately enclosed unit on its own. theres small screws holding the cover in place of the cd/dvd unit, and once those are taken off, you can access the lens itself.
Use either a really clean and new 100% cotton type of cloth, or better yet, a new eyeglass cleaner cloth. You don\'t want to end up depositing anymore residue on the cd lens. I just simply wiped the lens with a sock, and then put everything back together, and tada, my ps2 worked just as it did when it came out of the box. When games were played, the noise of the lens trying to read the media became less, and I had no problems loading any kind of games.
p.s. Another method I would\'ve tried is using one of those blowing duster thingies. However I didn\'t have any, and often times, its not only dust that\'s on the lens, but a thin film of residue and dust and stuff like that. Just think of it as washing a car. If there was bird poop on your hood, just simply trying to whipe it with ostrich feathers won\'t take the residue off. Using a cd/dvd lens cleaner in a sense is like ostrich feathers, and it\'d take off simple dust, but not anykind of residue which could have depositted on the lens. Oh yeah, and when I refer to residue, I refer to stuff such as cigarette smoke, dust/moisture combinations, etc. Although its in an enclosed case itself, dust will end up in there no matter what.
good luck.