You are not the first person to think of this... Peter Watts uses thid kind of oxygen source in his underwater fiction, including Starfish (full story available on his website for free) and in a short story snippet.
The rifters in Starfish had a lung replaced by a system that does the appropriate gas exchange itself, removing all the problems of bad things dissolving into and out of bodily fluids under extreme pressure, and the latter has a simpler system that produces breathing gas that is inhaled by an unmodified human.
I can't tell which system you're specifically interested in, though note that the former is more invasive and needs a direct gas exchange system with blood pumped through it (a little like ECMO, perhaps combined with liquid breathing, avoiding any high-pressure gas stages) to get oxygen in and CO2 out. The latter system uses the diver's lungs for this purpose (effectively hydrox diving). That's obviously simpler and uninvasive, but doesn't solve problems of gas narcosis or the bends. I won't go into further detail here, but obviously there are details you need to care about.
Lets consider your design ideas though:
I've elected to place them in large "wings" on their backs
A professional aerobic athlete might be breathing at 150 litres of air per minute at sea level whilst working hard, and this seems like an OK "maximum output" level. About 5% of each lungful is consumed, giving you a peak O2 flow of only an eighth of a litre per second.
The density of oxygen at STP is about 0.04013 mol/dm3 (a little less than the equivalent ideal gas), so we can see that our athlete needs ~5mmol of oxygen per second.
There's a single oxygen atom per water molecule. We therefore need 10mmol of water to general 5mmol of O2. The molecular weight of water is 18.01528 g/mol, therefore you can get your entire oxygen needs from .18g of pure water per second... an equivalent flow of just a 10ml per minute! (and if you're interested in checking my results this chemistry.SE answer will help)
Obviously flow rates in a real device would likely be higher, because you don't want your electrolysis cell to go completely dry. This avoids clogging from dissolved materials like salt or calcium carbonates, and stops liberated gas from getting too hot.
You can see that you can supply all the water you need for your electrolysing-gill needs from a tiny pump, with a tiny inlet hole (or several holes to reduce the risk of blockage). Similarly, a small vent hole (or series of holes) would be needed to vent any excess gas, which will be 10mmol/s of H2 if you're directly oxygenating the diver's blood.
A resting human needs more like 5-8l/m of air, which is vastly less than the athlete, and so someone floating or paddling gently will need an order of magnitude less water.
Note: if you are generating gas to be inhaled you will need to split more water to provide enough hydrogen to dilute the oxygen, because breathing to high a partial pressure of oxygen results in acute oxygen toxicity which can be fatal. This means you'll generate an excess of oxygen that will have to be vented into the sea, instead of an excess of hydrogen. Remember that whilst hydrox is potentially dangerous under normal circumstances, you're neither generating nor storing large quantities of the stuff, and what you do generate is either consumed immediately or vented. It should let you dive safely to depths of over 200m without narcosis risks. careful decompression will still be required, but this could be made be safer and easier than heliox diving.
There's no need for huge surface areas for extracting water by electrolysis.
And apparently saltwater is a good conductor?
It is, but we've established that your electrolyser can be pretty small and hence well protected. If necessary it could be run in a pulsed mode of operation where the water being electrolysed is electrically insulated from the outside by valves, and when waste hydrogen is burped out, water can be sucked in to refill the chamber. Electroceptive species like sharks might notice your device running, but whether they'd find it interesting or unpleasant I couldn't say.
Power requirements are a slightly larger problem. Water needs 237.24 kJ/mol to split, so to split the required rate of 10mmol water per second you need a ~2380W power supply, which at the necessary 1.23v equates to a nearly 2KA current. This is equivalent to the power supply needed for welding. You'll be wanting a decent fuel cell or battery pack to drive your air supply!
Of course, a normal person might need as little as a tenth of the pro-athlete's flow rate, requiring a reasonable 238W power supply, more like an electric bicycle which is obviously acheivable with modern day battery tech without being too bulky or expensive. Even the athlete can't sustain sprinting levels of effort for long, compared to the likely length of a dive.
In any case, future fuel cells should give you pretty good energy density, and future advances in artificial photosynthesis should provide the fuel which can be conveniently made at sea on the surface.
Some further chemical cleverness will doubtless be required. For example, electrolysis of salt water can release chlorine gas which you probably don't want to breathe, and chloride ions can damage the electrolysis equipment. Some research already exists in this area, though it is worth considering that some kinds of damage (or sabotage!) to your electolytic lungs in seawater could still cause chlorine to be released.
