Source for Colored String / Twine ?

I need something thicker than common threat, more along the thickness / weight of kite string or yoyo string.

Any fabric store would have a large selection of things like that in their notions section to be used as decorative trim. Something like you describe is probably going to be 30-40 cents/yard though, so it's probably too expensive to be suitable.

If it needs to be thicker for strength purposes, you can use upholstery thread. It is very similar to normal thread but can hold around 20lbs. It doesn't have a huge colour selection though.

Your best bet is probably to colour it yourself. Make sure you get a 100% cotton string; no polyester or other synthetic fibers. Then use RIT dye. Just mix it according to the package instructions (or up to double strength for darker colours), and submerge the string in it for a while. Rinse it very well in clean water, let it dry, and you're done.