This is described in the source at http://ftp.snt.utwente.nl/pub/software/tex/macros/latex/contrib/eqexam/eqexam.dtx.
\DescribeMacro{\examNameLabel} provides a line for the student to
enter his/her name into the exam. The command \cs{examNameLabel} can
be used to define the name label, the default is \texttt{Name:}
Since "Name:" is typeset while \output is active, macro \c@page (which outputs the current page number) gives the correct value in all instances. I suspect adding the line
\makeatletter\examNameLabel{\ifnum\c@page=\@ne Name:\fi}\makeatother
to your document, will suffice. This does not get rid of the \hline underneath, though, but you have not specified whether you want to keep that.
If you also want to get rid of the line on all subsequent pages, you will need to add the lines
\makeatletter
\examNameLabel{\ifnum\c@page=\@ne Name:\fi}
\eqExamName[\Ff\FfRequired]{\ifnum\c@page=\@ne2.25in\else\z@\fi}
\rheadeqe{\ifnum\c@page>\@ne\vrule\@width2.25in\@height\z@\@depth\z@\fi}
\makeatother
which also retains original spacing. A full MWE:
\documentclass[a4paper,12pt]{scrartcl}
\usepackage[myconfig,nosolutions,pointsonright,totalsonright,noparttotals]{eqexam}
\makeatletter
\examNameLabel{\ifnum\c@page=\@ne Name:\fi}
\eqExamName[\Ff\FfRequired]{\ifnum\c@page=\@ne2.25in\else\z@\fi}
\rheadeqe{\ifnum\c@page>\@ne\vrule\@width2.25in\@height\z@\@depth\z@\fi}
\makeatother
\title[T1]{Semestertest}
\author{John Doe}
\subject[FDPH]{FD Physics I}
\date{Spring 2015}
\begin{document}
\maketitle
\begin{exam}[Teil 1]{Part1}
\begin{instructions}[Teil 1]
Instructions\dots
\end{instructions}
\newpage
\begin{problem}[5]
Problem
\begin{solution}[.5in]
Solution
\end{solution}
\end{problem}
\end{exam}
\end{document}
You can check the spacing by commenting and uncommenting the added code.
\documentclass{...}and ending with\end{document}. – LaRiFaRi Aug 20 '15 at 09:41