I need the name of this one "component"
What you are looking at is a rubber dome switch. These are typically custom parts, and the minimum order quantity (and design costs) are likely to be out of your price range.

Depending on your requirements, you have a couple of options:


Some rubber dome switches are available in standard PCB-mountable packages (containing both the rubber dome and the switch contacts). The sizes and shapes will be limited, though.
Reuse parts from an existing rubber dome switch. For instance, if you needed one just like the one in the picture, you could cannibalize the switches from the original controller. If you needed to position the switches differently, you could cut apart the rubber between the domes and glue the resulting parts in place.
Consider using a tactile dome switch instead. These switches use a buckling metal dome instead of a rubber part. The metal dome is a reasonably standard part, and can be positioned on the board with adhesive tape. One caveat is that these switches are much "clickier" than rubber domes.
If you just need to detect a finger touch without pressure, you could use capacitive touch sensing. This will require a conductive surface on the switch.