In the BuildPartList you can set Allow Pickup Build Part then you can set the Pickup Items, these are the items that the player will receive when the pickup a build part.

To pickup a build part the player must be the owner of the build part, then if the build part can be picked up then you can use the hammer right mouse menu while looking at the build part and select pickup.

Below is an example of the Cooler, it is set to allow pickup and when the player picks up the cooler they will receive the Cooler item.

To select an item in the pickup items, click the + button, then set the Data Table to the MasterItemList. You can now use the Row Name to select any item you would like the player to receive when they pickup the Build Part.

You can also add multiple items by clicking the + again and follow the previous instructions.