Well, for starters, I am not sure what sort of answer you are looking for. If you just want to talk about the wire, then you can open a materials database and find a stronger material, or, even simpler, choose a thicker wire. Just make an estimation of the force required to break it, and select a material and cross section that give you, let's say, twice the tensile stiffness. Using two wires will probably up the cost though.
In case you are looking for something more complicated, here's some things that may help you out:
Is this meant for one fence, or is the project about a potentially mass-produced product? This is important for your cost.
In theory there are a lot of combinations that could give you the desired result, but you need to build an argumentation when you suggest a solution.
You have 1 objective and 1 constraint:
Fence is strong enough (<---- Constraint, this must always be satisfied)
Minimal Cost (<----- Objective)
Think simple. The above means that out of many possible solutions that satisfy the constraint, you want the most cost effective one.
My knowledge as far as fences are concerned is not much, so I can't really tell you what to do, but I can tell you what you need:
1. Mechanical calculations. Every solution that you suggest must have a background of calculations that prove or indicate that it will work as desired.
One thing that will really help you is to identify the factors that make existing fences break. Is it just the tensile strength of the wire? From experience, this has to do with the stiffness of the entire structure, so if you change the weaving (e.g. make the holes smaller) you can reduce the moments generated. Also, as I pointed out before, the way they are connected is also important.
2. Information about the cost determining factors. This will allow you to shape arguments on whether the solution you found is cost effective or not.
Examples of this would be the cost of acquiring machinery, the cost of electricity, man-hours, material costs (in case of a thicker wire, or a different material) and so on.