Electrolysing water makes hydrogen and oxygen, in the proportions 2H2O = O2 + 2H2
Doing that takes power (Duh!) which you can get back later as heat by burning it, or as electricity by shoving it through a fuel cell. The usual losses apply (it's about 70% efficient), the Universe won't give us something for nothing, it won't even let us break even.
The kicker when using it to power a mobile device is, I understand, the problem of carrying enough around with you to burn back to H2O and to make enough electricity to cart one's backside from A to B. I believe that some of the latest pressure vessels for hydrogen fuel-cell cars will take 800 bar, which gets round the problem of having to spend energy constantly refrigerating the hydrogen so it doesn't boil off - at that pressure, LH2 would boil at over 60 deg C. On the other hand, compressors take a hell of a lot of power in the first place.
Personally, I like the idea of STP direct methanol fuel cells, as used in the latest marks of air-independent submarines from HDW.