Spoke tension does not affect wheel stiffness unless the spokes go completely slack.
Things that do affect wheel stiffness:
A strong rim is stiffer than a weak rim. They used to make crazy light 280 gram alloy tubular rims back in the day, but they were so weak that you had to build them in 32 and 36 hole, which took away all the benefit of the weight savings. Nowadays rims can be fairly lightweight and strong allowing lower spoke counts to be used.
Spoke count does affect wheel stiffness though. More spokes will make a stiffer wheel and heavier riders are going to want to use a wheel with more spokes. Spoke type and thickness will also contribute to stiffness but can affect both weight and aerodynamics.
Hub flange spacing. Wider flanges make for a larger bracing angle. If you think to a triangle shape, a triangle with a wide base is a sturdier structure than a tall narrow triangle. Having the flanges spaced out as far as possible makes for a stiffer wheel.
Hub stability gets overlooked, and I wrote a blog about that recently:
http://blog.boydcycling.com/2013/05/hub-stability.html
And definitely do check to make sure that your hub is tight and also that your skewers are tight. If your hub is moving back and forth on the axle or if your wheel is moving around in the drop outs that won't be good for anything.