Can someone's scuba gear be improved (details inside) ?

They already do rebreathers

Electrolysis to liberate O2 in breathable amounts from water while exerting yourself would require some heavy batteries and tech that I am not sure is available at this time. Sounds cool, but it it requires some ampage, and you are wearing less than a 3 mil suit, it may turn out like dropping a hair dryer in a rather large bathtub. Then if there is a short in a pure O2 environment, look for anything combustible near the spark. Aside from those shortcomings, which can be overcome, all it takes is money.

Scuba divers don’t breathe oxygen, they breathe air. Extracting air from salt water is a lot harder than extracting oxygen.

As @majorrich pointed out, it takes quite a bit of electricity to crack water and get oxygen out of it; more power than any conveniently-sized battery has. If you replaced the ballast weights that some divers use to achieve neutral buoyancy with batteries, you might get enough oxygen for a breath or two… assuming you use the most energy-dense batteries currently in existence as opposed to something that is actually commercially available.

Also, you really don’t want to breath pure oxygen at depths much over about 5m; once you get below 6m, the partial pressure of oxygen would be high enough to cause oxygen toxicity. Of course, you probably won’t die from that immediately; going into seizures and spitting out your mouthpiece generally leads to drowning faster than oxygen toxicity would kill you.

Deep divers bottles with oxygen levels too low to support/sustain human life at normal atmospheric pressure, but at greater depths, it is enough to keep a person alive. But like many other things about diving, a lot depends on the depths you plan to go, how long you plan to stay there, and your own physical condition. There are reasons that divers have such extensive training before they are allowed to strap on some tanks and go for a swim.

Many of the limits imposed on divers are not technical though; they are biological. Some of those restrictions are based on the fact that they diver will probably eventually want to resurface. Often, limits on the lengths and depth of dive are determined more by what it would take to allow a diver to survive the resurfacing process than anything dealing with how much oxygen they have available.

@Judi Rebreathers are a whole different thing, mainly used to eliminate the telltale bubbles that would tell a sentry that SEAL Team 6 just swam into your place or spook the local fish. (Sharks don’t always take well to being startled!) They do allow you recycle your exhaled gas and extend the duration of your air supply, but they do not work by electrolysis.

@zenvelo It depends on the depth, but compressed air is far more common than pure oxygen, especially as most divers are recreational divers that don’t go down beyond the safe depth of air. Those that go deeper will run into serious issues with nitrogen bubbles in the bloodstream though, which is part of why so many deep divers use a lot of helium in their mix, whether Heliox, Heliair, Trimix or other blend.