When I did rock climbing I used spectra for slings, and for the loops on hexcentrics and nuts. It's very strong, but not noted durability, I think. It's made of a sort of polyethylene, oriented as I recall, so it's got a lower melting point than nylon. I also use much thinner spectra for kite lines for strength and low stretch - more like string. It is very strong for its weight, and has low stretch, which you want for large stunt kites.
If you go to a decent outdoor/climbing/backpacking store that sells ropes you'll see an assortment of nylon cordage ranging from shoestring replacement stuff to 11mm rope or bigger. I think any of the 3 to 5mm nylon stuff would be fine, and probably much cheaper than spectra, which, as I recall, was pretty pricy! They usually sell the small stuff in a few colorful patterns for making lanyards and tying stuff together. It's usually kernmantle construction, which means you have a braided outer core around either a braided inner core or a bundle of twisted strands. Either way, it takes knots fairly well but isn't really splicable.
If you don't care about color, go to an army navy store and get paracord. About 3mm or so, nylon, usually sold in 50 or 100ft hanks - more than enough for lots of lanyards.
If you want to get fancy, and don't mind using 1/4" stuff or larger, go to a boating store and pick up some dacron line and a 'fid' kit. You can splice eyes in the end of the rope, even splice it onto the object you're putting the lanyard on - no knots that way. Kind of a pain, but a really nice lanyard when you're done.