Crypto that takes n days to solve depending on how many bytes are missing? Is there a good candidate for an algorithm that would takes n days to break depending on how many bytes of the key are missing? I'm writing a story where you need to collect pieces (bytes) of a key and depending on how you have you get an head start in brute-forcing the message. 

The message is around 200 chars long (so quite small).  Count on computer resources be what would feasible for a non state-actor (bot nets etc), but still pretty resourceful. The key length / piece length does matter much, but should be viable for 6-8 "pieces"

The main problem I see is that time with brute forcing strong encryption is usually measured in years, or even million of years, and I need it more to be around 3 days per "piece"
You can try 20 billion DES keys per second on a single Nvidia Geforce GTX 1080.

Assuming (roughly) that another algorithm (with much higher keylength than 56-bit) is equally as fast, the bitlength you want for the key is:

$$\log_2(\text{NumberOfSeconds})+\log_2(\text{NumberOfGPUs})+34$$

where $2^{34}\approx 16 \text{ billion}\approx 20\text{ billion}$.

If you want some example numbers, let's do the math:
If we want to wait approximately 3 days (that's $24\times 60^2\times 3$ seconds) and can get hands on maybe the equivalent of 5000 such GPUs using a botnet, that yields about $64$ bits which sounds very reasonable.

What you could do with this is to take AES-128 und give 8 bytes using (8?) puzzles and let them brute-force the remaining 8 bytes. Every unknown byte will increase the brute-force workload by 256.

So TL;DR: Brute-force of 8 bytes (give or take a few bits) sounds like what you're aiming for.