A reasonable approach is probably to take a lightweight block cipher, $E_\cdot(\cdot)$, fix a constant $c$, and use the function $f(k) = E_k(c)$ as the function. While we have no proof that $f$ is a one-way function, it is a good candidate for a one-way function. In particular, if the block cipher is secure, then $f$ will be a one-way function.
There are a number of designs of lightweight block ciphers. As a starting point, you might look at Piccolo and PRINCE, Simon, LBlock, PRESENT, µ2. There might be better designs.
Another plausible approach is to use a lightweight stream cipher. Any secure stream cipher will also be a one-way function. For instance, you could look at using A5/1 or ACORN.