In his book Concise Inorganic Chemistry, J.D. Lee writes:
The production of $\ce{H2}$ from $\ce{H+}$ at a lead electrode is kinetically unfavorable, so a much larger potential is required than the standard electrode potential.
However the book doesn't state any reason. Is it perhaps because of the formation of oxide film? Also, if lead behaves as such, how was its standard reduction potential of $\pu{-0.13 V}$ found out? I have searched the internet and cannot find any definite answers.
When considering reduction of hydrogen, you should remember that hydrogen in gas phase exists as two-atom molecules and isolated hydrogen atom is really unstable comparing with most other forms. However, in electrolysis of water reduction proceeds as a one-electron transfer and to make formation of molecular hydrogen possible, atomic hydrogen must stay somewhere waiting for a partner to form a molecule. Furthermore, formation of this molecule and subsequent desorption should be possible, both in a sense of a reasonable barrier and an overall acceptable entahlpy of the reaction.
As an example, copper is well known for its high overpotential in electrochemical hydrogen reduction, and this is commonly explained by its inability to adsorb hydrogen. Platinum is well known to be an effecient catalyst of many hydrogenation reactions and is well known to have minimal overpotential.
Lead is a main-group heavy metal, meaning that it forms fairly weak bonds with hydrogen due to very large and diffuse orbitals. Their further delocalization in metal makes adsorption of hydrogen on lead surface even more unfavorable.
For further info on topic, a good search term for Google Scholar would be "Electrocatalysis of hydrogen evolution reaction" (beware autocorrector mangling the first word).
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