The algorithm for placing sub- and superscripts is "hard-wired" in the TeX engine, and it's not as simple as you might think. First of all, there's a special treatment for sub- and superscripts of "simple characters" like a, \alpha or \mathcal{A}. In this case you also have some horizontal displacement of the subscript that is determined by the italic correction of the character: for example, the subscript should be shifted to the left for an F, but not for an A.
The default values for raising and lowering are as follows in the case of "simple characters". The superscript is (tentatively!) raised by the maximum of (depth of superscript) + (1/4 of the x-height) and σk, where the x-height is the \fontdimen parameter number 5, and σk is \fontdimen parameter number k (of \textfont2 unless the whole formula is in a sub- or superscript itself). Now it depends on the current style (which, unfortunately, the TeX user can only access via \mathpalette or \mathchoice): In \displaystyle k=13 is used, in "cramped style" k=15, and k=14 otherwise.
How much the subscript is lowered depends on the presence of a superscript. If there's a superscript, then the subscript is (tentatively!) lowered by σ17, otherwise by the maximum of (height of subscript) - (80% of the x-height) and σ16. In the case of a sub- and superscript combination there's an additional adjustment (with some unpleasant side effects), see e.g. this answer of mine.
If the sub- or superscript is not attached to a "simple character" but to some other nucleus, then additional minimum shift values are used: the superscript is raised by at least (height of nucleus) - σ18, the subscript is lowered by at least (depth of nucleus) + σ19, where σ18 and σ19 are to be taken from \scriptfont2 unless the whole formula is in a sub- or superscript itself (in which case \scriptscriptfont2 should be used).
I'm not sure if you really want to implement this ... if you do, have a look at Rule 18 of Appendix G in the TeXbook for details.
Best Answer
Type 'H', then Insert > Formatting > Subscript, then press the right arrow (this gets you out of the subscript), then type '0'.
You can assign this operation to a key shortcut such as [Ctrl]+[,] using
script-insert subscript(see also: lyx shortcut for superscript and subscript).