\documentclass{beamer}
\beamertemplatenavigationsymbolsempty
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{frame}[fragile,t]
\frametitle{1}
\begin{tikzpicture}[
dot/.style = {circle, fill, node contents={}},
every pin/.append style = {pin distance=9mm, pin edge={black, semithick},
text width=24mm, fill=orange!20}
]
\draw [thick,-latex](0,0) -- (6.2,0) node [black, xshift=.3cm, yshift=0cm] {$x$};
\draw [thick,-latex](0,0) -- (0,5) node [black, xshift=0cm, yshift=.3cm] {$u$};
\draw [ultra thick,blue] (0,2.84) --
node [dot,pin=60:very very very very very very very long long long long long long long text] + (-40:4.44cm);
\end{tikzpicture}
\end{frame}
\end{document}
gives:

is this what you looking for?
in mwe above i define node style dot for black circle on curve, text is added as pin to this node- pin style among others contain text width=<width> option. this option automatically break long text into more lines (long as determined in <width>)
addedndum:
more control on pin and text anchors enables solutions provided in Qrrbrbirlbel answer on the question tikz-adjusting-pin-anchor. adoption to your question is:
\documentclass{beamer}
\beamertemplatenavigationsymbolsempty
\usepackage{tikz}
\begin{document}
\begin{frame}[fragile,t]
\frametitle{1}
\begin{tikzpicture}[
dot/.style = {circle, fill, outer sep=0pt, node contents={}},
aligned pin/.style args={[#1]#2:#3}%
{pin={[pin distance=9mm, pin edge={black, semithick, shorten >=-8pt},
inner sep=0pt, label={[append after command={%
node[text width=32mm, fill=orange!20, outer sep=0pt,%
at=(\tikzlastnode.#2),% position of pin
anchor=#1,% for node withpin text
]{#3}}]center:{}}% for details see https://tex.stackexchange.com/questions/246995/
]#2:{}}%
}
]
\draw [thick,-latex](0,0) -- (6.2,0) node [black, xshift=.3cm, yshift=0cm] {$x$};
\draw [thick,-latex](0,0) -- (0,5) node [black, xshift=0cm, yshift=.3cm] {$u$};
\draw [ultra thick,blue] (0,2.84) --
node [dot,aligned pin={[west]45:very very very very very very very long long long long long long long text}] + (-40:4.44cm);
\end{tikzpicture}
\end{frame}
\end{document}
which gives:
