Get a quality wax and oil leather dressing, like Fiebing's. Don't use neatsfoot oil, and don't saturate the leather. Warm the sling slightly in an oven, on the lowest setting (the leather should not be too hot to touch, just warm). Apply a thin coat of leather conditioner, and the warm leather will drink in the conditioner, leaving a protective layer of beeswax on top. It will darken the leather, so be prepared for this. Several thin coats are better than one heavy, saturating coat, if the first application doesn't work. If the conditioner can't penetrate, the leather needs a very light, gentle cleaning with glycerine saddle soap. You might want to do that first, in any case. Just follow the directions on the bottle.
Cheap leather is cheap leather, though, and it may not ever feel exactly like you wanted. Even top quality leather will benefit from proper care, and occasional leather conditioning. It dries out, like your skin can, and that lost moisture needs to be replaced from time to time.
