I feel like this could be more useful for cleaning up contamination than for electricity generation. Is there a comparison of how much energy is used to capture the uranium vs how much energy can be obtained from that amount? If there's no net gain, it seems best for cleanup duty, but with a useful waste product.

The vacuum of “empty space” appears to be an energy reservoir of immense capacity that makes nuclear power seem old fashioned. Richard Feynman and others have pointed out that the amount of Zero Point Energy in one cubic centimeter of the vacuum (ambient space) is much greater than the energy density in an atomic nucleus.
GE’s Vision for the Superconductor Industry (stated in 2002) - "Electric generators made with superconducting wire are far more efficient than conventional generators wound with copper wire. In fact, their efficiency is above 99%, and their size about half that of conventional generators … ” http://www.superconductors.org/uses.htm

If wind power is used for the electricity needed in this system, and possibly hydro-power it could help to reduce the expense of the power costs. Maybe a centrifuge type system to enhance the collection of uranyl after gathering on the fibers.

We read, "For much of this century, some fraction of our electricity will need to come from sources that we can turn on and off," says Steven Chu, co-author of the study. "I believe nuclear power should be part of that mix,..." First off nuclear power can't be switch on and off as suggested. Second, nuclear is too expensive compared to renewables combined with pumped hydro or other means of storage. Third, what is the effect of this process, the "jolts of electricity," on life in the ocean. What percentage of the ocean would need to be processed this way to harvest a meaningful amount. Civilian nuclear power is dead, so why is former Secretary of Energy Chu clinging onto it.

This is Insanity. So... you want to take low level material thats spread out all over the place in light concentrations... and then capture it into a super-concentrated form, where by oceanic creatures would be fried... meanwhile, the device could easily fail, and be unrecoverable... wreaking catastrophic damages on a global scale.
These Scientists are Out of Control. Absolutely out of their minds.
Even in a "Clean Up" format, which is SEVERELY needed for Fukushima... and probably the continued existence of humanity... This method could be extremely dangerous, and cause more damage than it tries to solve.
There had better be some high level "Devils Advocates" for any attempt at design / builds. So far, Man's track record of making things better... has resulted in some of the most catastrophic and negatively impacting pollution/destruction ... that this world has ever seen.

Or could it extract mercury from ocean?

I always thought that was the point of fast breeder reactors. You don't need fuel once you start processing your waste

Cleaning the rapidly increasing amount of water that is being stored in tanks at Fukushima makes a great deal of sense. It will still take a long time to get to the point where water contaminated today can be re-used to cool Fukushima tomorrow. Same thing at Chernobyl Ukraine, Hanford in Washington state, Oak Ridge National labs in Tennesee, Gathering uranyl from the ocean seems a lot less likely to ever be a viable idea. None the less I would prefer to have tools and choices that MAY be useful rather than do little or nothing except rail at an unfortunate disaster.

Uranium is very depressed right now. I am still waiting for my Uranium Resource stock to come back. Fuel is a very small proportion of cost of nuclear energy. The value of naval reactors is that they can be turned on in seconds, in the event that an ICBM is headed for the carrier.

Filtering uranium from seawater -has- to be cheaper than mining it from high-content ore in the ground, right? <g> Yes, where possible, use this process for cleanup, and if possible, find a way to modify it for mercury containment in seawater. I agree with Chu and would rather have nuclear power generation (yes, which can be switched in and out of the grid at will, but the fuel never turns off) than dirty, uranium- and CO/CO2-spewing coal. Go modular for reduced regulation cost. Recycle fuel for far less nuclear waste.