I agree with the previous answer that there are a number of issues with implementing a serial hybrid (e.g., a chemically powered generator running electric motor(s)) on an aircraft, but I think there are a couple of positive sides to it as well...though we may have to design a new propulsion system rather than supplement an existing one. I found a NASA study looking at distributed turboelectric propulsion to replace cryogenically stored hydrogen scramjets on a BWB aircraft which shows the potential to actually save weight by moving over to the electric system (https://mdao.grc.nasa.gov/publications/IPLF08-Kim.pdf), but that's a pretty niche application. This, naturally, does not universally imply that that is the case, however: just for now and for that application.
However, the mention of distributed propulsion does highlight a major plus for electric propulsion systems. It's dramatically easier to route flexible power lines than mechanical shaft connections. Hence, if you want to have multiple engines...electric propulsion enables you to do that relatively easily. Also, say that you want to make an aircraft that can hover and cruise as a typical fixed wing aircraft -- typically, you'd have to design collective pitch change into your proprotors. Or, with electric propulsion, you could just have a bunch (I mean 12+) of proprotors, each of which you tailor with a twist distribution proper to a certain mode of flight. Look at a proposed NASA design (http://aero.larc.nasa.gov/files/2012/11/Distributed-Electric-Propulsion-Aircraft.pdf), a separate design by Aurora Flight Sciences that has advanced in the DARPA X-Plane program (http://www.darpa.mil/news-events/2016-03-03), or Lilium Aviation's recent design (http://lilium-aviation.com/). Given all of these are concepts, so nothing is proven by a working design as of yet...but it seems like a lot of people are looking into the possibilities afforded by a distributed electrical propulsion system. The expected advent of lithium-sulfur batteries should be a big deal here too. These batteries are expected to have a capacity of ~500 kWh/kg (about 2x the absolute best Li-poly available now), but I think that 2019 is the expected date for high-capacity Li-S batteries to start hitting the market.
Finally, one thing about engines is that their efficiency can vary dramatically depending on how they are being run (i.e., the load on the engine, RPM, etc.). This excerpt from a German textbook is, unfortunately, in German (https://books.google.com/books?id=QAGHZPVnnSAC&pg=PA540#v=onepage&q&f=false), but the basic idea is reflected on a Wikipedia page where the plot is reproduced in English (https://en.wikipedia.org/wiki/Consumption_map). On the vertical axis is power output of a spark-ignition engine, the horizontal axis is the RPM that the motor is being run at, the the contours are, essentially, fuel consumption. The long and the short of it is that, if you have any change in the engine's load or what RPM it's running at, it's not running at its most efficient. This would be the case for your typical non-hybrid propulsion, where load and RPM vary with flight condition. However, if we decouple flight condition (i.e., desired cruise speed, throttle setting, prop speed setting, prop pitch setting, etc.) from the chemical engine and let an electric motor (which has a much higher efficiency than a chemical engine) deal with these variations, we can, conceivably, run the chemical engine within an optimal performance range (some variation will probably be needed if power output is excessive, but that's a design problem wherein you'd try to size the engine for its most efficient in cruise, I'd think). That's a fuel savings for us, plus a decrease on engine wear because it doesn't have to see all the throttle variations typically present in normal flight.
As noted, this isn't the case for a typical airplane (small RPM bands)...but what if we decided to play the game that Aurora, Lilium, and NASA are playing and make VTOL/STOVL aircraft? The differences between flying a proprotor in edgewise and axial flight are dramatic, and having RPM control can make a lot of difference and might obviate a need for variable pitch propellers. Just design your twist distribution right, vary your RPM, and you just might be able to get acceptable efficiency in both regimes (at least for slower speeds). Or, like NASA, use two different sets of proprotors for a fixed-wing cruise and vertical flight, each one optimized for that specific flight regime. I'm not saying it's easy...just that electric propulsion makes the opportunity present.
But, yes, weight and technology are major concerns...and you can see in the electric aircraft that have been built to date. Firefly (Sikorsky's electric helicopter) has a max endurance, I think, of 15 minutes. Helios (https://www.nasa.gov/centers/armstrong/news/FactSheets/FS-068-DFRC.html) is a neat aircraft, but look at how flimsy the design is in order to enable it work as a solar/fuel-cell hybrid (i.e., solar just doesn't provide a lot of power, so you need to make a light, high-aspect ratio aircraft). The Gamera-S is a solar-powered quadcopter under development at the University of Maryland (see below), but look at how sparse the frame is--weight is a HUGE problem for these aircraft, especially given how much power we can currently get from batteries or the sun. Serial hybrids don't get rid of that problem (since it introduces engine weight, fuel weight, lubrication system weight, etc.) but it can give you a couple of perks, as noted above.