I would pick Gridcoin (GRC) as it is one of the more interesting coins out there.
Gridcoin (GRC) replaces mining with doing research for scientific projects, for example to cure Ebola (
http://www.worldcommunitygrid.org/research/oet1/overview.do), fold proteins (
https://boinc.bakerlab.org/) or find aliens (
http://setiathome.ssl.berkeley.edu/). Then the money is worth the useful computations that were given to the public to create the money.
Making this possible in a dencetralized way is very hard, but I think the current approach of Gridcoin solves this problem very well. You can read about it here:
https://docs.google.com/document/d/1PE4Rb6JylkVxj031pHW6aNPU-PLh_VenLHTkgqcZ2NM/edit#heading=h.1nx0jl77hp5rFrom a user's point of view he simply has to run the simulations from the different projects on BOINC and the Gridcoin client. Then Gridcoin comes to a consensus, who has contributed how much computing power to BOINC, which can bee seen for example here (
http://www.gridresearchcorp.com/gridcoin/?result&t=CPID_Leaderboard). In some blocks this consenus is saved in the blockchain and the reward one gets when staking a block is proportional to how much research was recently done for BOINC by this researcher.
More information and tutorial to set it up here: gridcoin.us