BibTeX: Abbreviate first name (aka given name) to 2 or 3 letters (not 1)

Just replace the first name in question, say Csaba, with {\relax Cs}aba.

More generally, place the part of the name that is to be unaffected by the abbreviation inside a TeX group, which is a field that's delimited by { and }, and insert the command \relax at the beginning of that group.

Another example of how to make use of this method: To help one's readers distinguish the works authored by "Timothy Jones" and "Thomas Jones", one could set the respective author fields to "{\relax Tim}othy Jones" and "{\relax Th}omas Jones".

Here's an MWE that suggests how to deal with four authors whose last name is "Kovács" and first names begin with "C".

\RequirePackage{filecontents}
\begin{filecontents}{references.bib}
@Book{kovacs_ca,
  author  = "Carla Kov{\'a}cs",
  title   = "Title 1",
  year    = 1986,
  publisher = "Addison-Wesley"
}
@Book{kovacs_ch,
  author  = "{\relax Ch}arles Kov{\'a}cs",
  title   = "Title 2",
  year    = 1987,
  publisher = "Addison-Wesley"
}
@Book{kovacs_cs,
  author  = "{\relax Cs}aba Kov{\'a}cs",
  title   = "Title 3",
  year    = 1988,
  publisher = "Addison-Wesley"
}
@Book{kovacs_cy,
  author  = "{\relax Cyr}ill Kov{\'a}cs",
  title   = "Title 4",
  year    = 1989,
  publisher = "Addison-Wesley"
}
\end{filecontents}

\documentclass{article}
\usepackage[T1]{fontenc}
\bibliographystyle{abbrv}

\begin{document}
\nocite{*}
\bibliography{references}
\end{document} 

Put

\newcommand{\Cs}{Cs}

in your .tex document and use

author="{\Cs}aba Kov{\'{a}}cs"

in the .bib file.

An automatic solution working with Biblatex and Biber can be found here: How to abbreviate first names with multiple letters using Biblatex?