Cord & Connectors
Creating the payload train by connecting the piece(s) of your mission to the flight line sounds simple enough, like building stairs, but when you really get into it you might find yourself in a bind. Some of the decisions will be reasoned out here, but others will still require your participation and ultimate decision.
Knots to Use
When you need to use a knot for securing any part of your payload train then you will want to use a self-tightening knot. Probably the most popular option, also used for rock climbing, is the figure-eight knot. Another option used for high altitude ballooning is the bowline knot, which is much easier to untie after it has been loaded (more important for something like rock climbing, but not irrelevant here).
Keeping Your Enclosure on the Line
Running the rope through the center of your payload box can work well. I have had students attach fender washers (more surface area for a given hole size, better for attaching to a soft surface like foamcore) to the outside top and bottom of their enclosure. The washers will ideally be just big enough for the rope to pass through, but not big enough for the knotted rope to pass through. If in doubt, get a few fender washers near the size that you think you need. Attach the largest of the fender washers to the box and then you can add/remove smaller washers outside of that until you get the kind of clearance that you want. Make sure the items on your payload train are not attached so close to the balloon that they are likely to hit it when experiencing severe turbulence.
The Great Regulatory Challenge
I have seen project after project err on the side of having rope that is very (too) strong, often Kevlar®- or Spectra®-based. This seems like a reasonable engineering decision, but I have sustained a concern about the FAR 101 regulations that require the break strength of the load line on an exempt launch to not exceed 50 lbs. I have never found a discussion about this online and have never gotten a straight answer in multiple conversations that I have had. I found a cord on McMcaster-Carr  that has a break strength of 20 lbs, which sounds reasonable and “exempt,” but it has a safety factor of 12 so it is well beyond the FAA requirements for an exempt launch. Key rings, car key carabiners, or even some rings of hook-and-loop fasteners  (Velcro®) could be reliable ways to attach very strong cord to reasonably strong attachment points that will satisfy the FAA requirements.
Many HAB enthusiasts believe that swivels are required equipment on the payload train to let out any twisting that can occur. On the other hand, I don’t know of any mission failures due to not including a swivel on the payload train. Heavy duty fishing swivels are likely the kind of solution that will take care of this problem economically.
Something to Discover
How do the cord or hook-and-loop fasteners respond to very cold temperatures when they are rated to just 0 °C?
 I have always said that if you cannot build something with the materials found at The Home Depot, DigiKey, and McMaster-Carr then you are up to something really crazy or you are doing it the hard way.