Has anybody ever tried sewing a weight on the end of the rope that is thrown towards the dock? Something maybe like a lacrosse ball.

How many times has somebody been standing on your platform, not able to throw the line and pull your transom into the dock??

I think these are called slungshots in your an old salt: http://en.wikipedia.org/wiki/Slungshot

If you had a long line, you could heave that thing a pretty long distance.

A tennis ball and very small line could be tossed less dangerously, tie the free end of the tiny line to the mooring line and let the tossee pull the heavy line over.

monkey fist knot.

That's all well in theory but when the current is ripping and the wind is blowing, I don't think there is usually time to throw the ball with the light line, then pull over your regular one.

I run 30' bow lines and 25' stern. 1/2" carries straighter and catches easier than 3/8 which seems to squirm more and slip though catcher's hands.
Very few missed catches plus easy, safe in/out splits of a raft line.

I also keep 100' of cheap diamond braid poly easily accessible, precisely coiled and bound with velcro for big gaps or emergency throws.
Eyes tied on both ends makes holding/cleating easy.

I don't have to deal with rolling seas, tides or current so even longer reaches may be helpful.
If more weight is needed, I would consider a small sandbag like used to hold down boat covers?

I also carry 30' lines but rarely is somebody capable of throwing it even 15' and if the wind is blowing in your face, forget about it. You need some mass on the end of the rope to carry it...

Rope cannon?!

I have heard that a cat works very well up to 50'. Have not tried it personally as I was concerned about the lacerations that might ensue.

brick? or is that a bit narrow?

You Midwest folk could use potatoes.......