Can I get a discount on my flight for not being overweight?

Samoa Air does, but the demographics make the reason for that obvious. http://www.telegraph.co.uk/travel/travelnews/10127347/Samoa-Air-introduces-XL-class-for-larger-passengers.html

Also an individual passenger's weight is not that important until we are on very small planes.

A big reason airlines want to keep baggage weight down isn't because the weight of the baggage (or passengers) directly increases fuel consumption (or other direct costs) but because baggage carriers have to move those bags. If everybody has bags that are unwieldy from weight or size then it will take a lot longer for the throwers to move the bags to/from the cargo bay. It doesn't do the baggage handlers any good that the passenger associated with a really heavy bag is slim.

Get a big coat, with lots of pockets.

No-one on a regular airline is going to give you a discount for being slim. You still occupy a whole seat, don't you? If you are so overweight that you need two seats, that's another story. They'll rightly charge you double. A while ago Ryanair charged triple for people who need two seats due to obesity or other reasons (such as having a leg in a cast.) I don't know if that's changed.

The price of excess baggage is just a huge moneyspinner. It doesn't really cost the airline $20 to fly an extra kilo, but they'll charge you for it if your bag is overweight.

Just get a coat, and stuff the pockets as full as you can. You can get 5kg in there if you have heavy stuff. And stick a book down the sleeve.

If you're carrying shoes and boots, wear the boots on the plane. You'll have to take them off at security, which is a hassle, but you'll save money.

Another trick is to put your stuff in a duty free bag. Generally the airports won't let the airlines charge extra for duty free, and no-one's going to check what's in there.

The answer given by reirab is a very good one. I'd like to add the aircraft operations perspective. For airlines, having passengers pay based on their weight is non-trivial to implement.

Before a flight, the airline must determine the total weight of the aircraft. This includes the weight of all people on board. The aircraft may be full of very skinny people, if the airline doesn't quantify how much lighter than are compared to average, then the airline can't optimize their operations accordingly (e.g. they can't reduce the amount of fuel on board or carry extra cargo).

If the aircraft has more than 10 seats, then the airline must either weigh each passenger individually with their hand luggage or use a forfeitary value. To give an example, during the winter on a plane with more than 30 seats, the weight of all passengers can be counted as

84 kgs x number of passengers

So for instance, if you weigh 45 kgs, the airline can't carry 39 kgs of extra cargo and they can't reduce the amount of fuel aboard. You not being overweight is not actually saving the airline any money.

If an airline wants to use the individual weights of passengers rather than the forfeitary value, then each passenger must be weighed individually before boarding the plane. This solution has been considered by a few low-cost airlines, but has always caused a public outcry. Furthermore, the average weight of the passengers+hand lugguage will likely be very close to the forfeitary weight.

In the case of Samoa Air that was mentionned in another answer, it is my understanding that they operate aircraft with less than 10 seats. Therefore, the forfeitary weight they must use is 104kgs per man and 86kgs per woman. In this case, using actual weight rather than forfeitary weight can make a difference. Furthermore, they can use passenger declarations rather than weigh each passenger, making it easier to implement.

They aren't charging you for your weight, they are charging you for the seat you are taking up. It doesn't matter if you weight 120kg, 70kg or 30kg, you are still going to take up one seat. So, except on rare occasion (like Somoa Air), they will never give you a discount for weighing less.