How difficult is it to produce diatomic hydrogen?
Diatomic hydrogen is hydrogen’s stable elemental modification under standard temperature and pressure — so if you generate hydrogen you will be generating diatomic hydrogen.

The simplest way to do this with household means is probably to use acetic acid concentrate (the stronger the better) and zinc powder. (Solid zinc works but much slower.) The reaction may be sluggish; if that is the case, very carefully and slowly heat the acetic acid. Capture the hydrogen in some glass, cup or test tube held upside down (hydrogen is the least dense gas with each molecule only weighing $2~\mathrm{u}$ (atomic mass units) so it will always rise).

If you have collected some amounts of hydrogen gas you can proove its presence. This was a common experiment in my high school chemistry courses so I’m going to tell you how to do it. However, exercise caution! If you didn’t do this at school yourself or didn’t watch someone do it before, I would advise you to not attempt it. Only do this with very small amounts of hydrogen, no more than a test tube of $2~\mathrm{cm}$ diameter. Make sure the test tube is full (by collecting enough hydrogen). Carefully bring the opening towards a candle flame. Do not use a pocket lighter where your thumb is right next to the flame for safety reasons. The hydrogen should burn with a faint ‘pop’ creating water. Considering how slurrish the reaction of zinc and acetic acid probably is, it may be too slow to actually collect enough hydrogen. Don’t attempt the burning if that is the case.

The reason for being cautious is the following: $2\,:\,1$ mixtures of hydrogen and oxygen are explosive. And with explosive I mean that they give a very loud bang. Luckily, air is mainly nitrogen so the probability of accidentally creating an explosive mixture is rather low. However, if you did, you will regret it.

Finally, on the topic of ‘hydrogen-enriched water’: Hydrogen, like all other gases, is not well soluble in water. However, it is hardly present in standard water since it is almost non-abundant in the atmosphere so it cannot dissolve. (Carbon dioxide, nitrogen and oxygen are far more abundant and therefore spontaneously dissolve in water to a certain extent — especially carbon dioxide which, in fact, reacts.) It is possible to create a hydrogen-saturated aquaeous solution or something less hydrogen-containing and sell it as ‘hydrogen-enriched water’. It would have no positive or negative effects on you or your surrounding. (Unless maybe your voice got slightly lighter in the same way as when you breathe helium — however, the amount of hydrogen is very likely too low for this to happen.) So it sounds like my dream of life: Making money with other people’s stupidity. It’s a shame that I’m not ruthless enough to do such a thing …