How can we implement a PRF with 1bit input space and 1024bit output space using AES securely?
It is secure. There would be an issue if that you generalize the idea to ridiculously long outputs. Any time you use two distinct inputs with a PRP using the same key, the outputs will necessarily be different. But for an ideal PRF there is a tiny chance that different 128-bit blocks could share the same value.

Because you only have 8 blocks, though, you wouldn't actually see duplicate blocks from an ideal black box PRF in practice. As a rule of thumb, you would not expect to see any duplicates until you look at around $2^{64}$ blocks. (See the "birthday paradox" and switching lemma.)

I'd discourage someone from doing anything more complicated with AES to mask that potential artifact. ECB encrypting the blocks with values 0 through 7, if the input is 0, or the values 8-15, if the input is 1, would be sufficient. 

A variable nonce isn't actually needed. You can just use a constant, such as zero.

Alternatively you could replace AES with a (non-block-cipher) PRF. HMAC or ChaCha20, for example.