What's the easiest/cheapest variable-frequency sine wave oscillator?

Wien bridge with a pot to vary the frequency. I bet you could build one for less than one US dollar.

Making a numerically controlled oscillator (NCO) with uC + DAC is very easy. Could be a fun FPGA project. An advantage to an NCO is that you change waveforms.

I did a low frequency numerically controller oscillator Arduino sketch (see http://wiblocks.com/docs/app-notes/nb1a-nco.html). At the bottom of the webpage are a couple of references to the original articles,

You didn't specify the frequency (100Hz or 100MHz?) or how much the frequency had to be varied (0.01% or 1000% ?) or whether the frequency had to be varied by a voltage or a physical knob. Purity of sine wave and stability matters too.

A one-transistor FET Hartley oscillator is hard to beat for cheap.