I'm not sure what you will find for under $120. That is pretty dirt cheap. But I could be wrong. If it were me, I would be expecting more in the neighborhood of $200-$300. Even for just the mattress---they often have less than a $50 difference to throw in the box springs.
I bought a foam topped full size set for my son this spring, for $399. Bought a very similar set (pillow top) for our daughter (also full size set) also for $399, including free delivery and carry away of the old stuff. But I took a good month or four shopping around. A twin set for our daughter 7 or 8 years ago cost us $300. And it wasn't a great mattress either. But, I didn't have much time to shop around for it--as we were GIVEN a twin bedroom SET for her (tall dresser, chest of drawers with mirror, night stand, poster head/foot boards and rails. The whole shebang--we just had to get a mattress, and we didn't get but a couple days notice that we were getting any of it!).
It wore out quickly and we should've replaced long before we finally did (last December). Got her a new (full sized) bed, too. :)
I know your little guy is small now, but trust me, he will grow. And quickly. If you go bottom of the barrel cheap, he will feel it by age 7 or 8. And while that seems like a long time away, it really isn't. A mattress should last a lot longer than 4-5 years. It will be worth it to spend a little more for something that will last him 8 or 9 years.
One other suggestion: buy a set of whatever sheets you intend to put on the bed and see how deep the pockets are. Most sheets sets have much deeper pockets than they used to have, and they won't stay on or fit properly on a too thin mattress. It is next to impossible to find sheets that are less deep than about 12-14 inches. Unless maybe if you get the "character" sheets. But he will outgrow those over time, too....
