If you're looking for a new Proof-of-Work, I can recommend the Equihash Proof-of-Work that we selected for Zcash.

We studied Proof-of-Work functions for a long time, chose Equihash (https://z.cash/blog/why-equihash.html), hired a legendary hacker to study it and write his evaluation (https://z.cash/blog/the-zcash-equihash-analysis.html), gave out $30,000 dollars in bounties to make the best implementations for CPU and GPU (https://z.cash/blog/announcing-miner-contest.html), got lots of good open source implementations (https://zcashminers.org/submissions), launched the network (https://z.cash/blog/zcash-begins.html), and mining has been working very well at scale ever since (http://www.coinwarz.com/network-hashrate-charts/zcash-network-hashrate-chart).

The reasons we chose Equihash are:

* it is memory-oriented rather than computation-oriented which makes it less cost-efficient to implement in ASIC,
* it is asymmetric, meaning that verifying a solution is much cheaper than generating that solution (_even_ starting from the nonce that generates that solution); In particular it requires substantial RAM (hundreds of MB, depending on parameter choices) to generate a solution (or an attempted solution), but it does not require that much RAM to verify a solution. This may be useful for constrained implementations such as SPV wallets in constrained hardware, implementation inside the Ethereum VM, etc.
* it has a good level of scientific investigation behind it, which makes me think it relatively unlikely than an algorithmic breakthrough would enable someone to find solutions much cheaper than the competition. This has been born out by the experience of live Zcash mining, where developers have made tremendous progress on micro-optimization, but as far as we know no algorithmic breakthroughs.

A reason to choose Equihash is that you benefit from the all of the research and implementation work described above. There are many well-tested implementations available, both open source and proprietary.