Why is the hydrogen electrode used as the reference to set electrode potentials? Could any other element be used instead?
There are two separate issues here. The first one is defining a zero for potential. In theoretical physics, it is the energy of two charges separated by infinity distance.
In electrochemistry, this is chosen to be the potential of the hydrogen electrode since that is something that can be made reproducibly and hydrogen was the de facto 'zero' for a number of other quantities.
These days, there are much more practical and easy to use reference electrodes like the silver/silver chloride (as chipbuster mentioned) or the saturated calomel or the mercury/mercurous sulfate. Here is a good list of them from a manufacturer of such electrodes: http://www.koslow.com/select_reference_probe
Full disclosure: I used to work for a company that resells electrodes from Koslow.
There are different approaching points here. One is history: The hydrogen electrode has been used forever and people are used to defining values relative to it, so why not continue?
A second point of approach is the following: You cannot measure absolute potentials; you need some kind of a reference. The choice was completely arbitrary. However, there are some ‘rationalisations’ (that probably only turned up after hydrogen was already used):
Hydrogen is the first element in the periodic system. It is nice to have ‘number one’ as your reference.
The potential of hydrogen (hydrogen gas at 1 bar in an acidic solution of pH 0) is approximately equivalent to that encountered in 1 molar acidic solutions (sulphuric, hydrochloric, nitric acid etc.). Probably even predating the selection of hydrogen, metals that dissolved in 1 molar acid were termed non-noble and those that resisted the acid were noble. Using the hydrogen potential as a zero-point, the non-noble metals have a negative standard potential while noble ones have a positive. An easy way to distinguish noble metals just by a number.
But just to restate: The choice of the electrode as a standard was likely before the second implication was fully realised.