How to control label positions in tikz-euclide

You could compute a point radially outside of the circle at each point to place the node:

The label for the origin is placed at a point opposite of the (B) node. A slightly better solution would be to compute the angle midway between (OA) and (OC) and place the origin label at a 180 degree offset from that.

Code:

\documentclass[tikz,border=2pt]{standalone}

\usepackage{tkz-euclide}
\usetkzobj{all} 

\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(2,1){O} % Defines the point at which the origin is located
    \tkzGetRandPointOn[circle=center O radius 1.5cm]{A} % Gets random point of the circle with center at O and radius 1.5cm
    \tkzDrawCircle(O,A) % Draws the circle
    \tkzDefPointBy[rotation=center O angle 100](A) % Defines a point...
    \tkzGetPoint{C} % named C
    \tkzDefPointBy[rotation=center O angle 78](A) % Defines a point...
    \tkzGetPoint{B} % named B
    \tkzDrawPoints(O,A,B,C) % Draws dots
    \tkzDrawSegments(C,B B,A A,O O,C) % Draws the segments

    % compute vectors of each point form the origin
    \coordinate (OA) at ($(A)-(O)$);
    \coordinate (OB) at ($(B)-(O)$);
    \coordinate (OC) at ($(C)-(O)$);

    \node [red] at ($(O)+1.15*(OA)$)  {A};
    \node [red] at ($(O)+1.15*(OB)$)  {B};
    \node [red] at ($(O)+1.15*(OC)$)  {C};
    \node [red] at ($(B)-1.15*(OB)$)  {O};
\end{tikzpicture}
\end{document}

I added a part of my answer in the next version of tkz-euclide.

\tkzAutoLabelPoints needs a point called **center** and a coefficient called **with**, this number represents  a percentage of the distance between the point and the center. By default it's 0.15

\documentclass[tikz,border=2pt]{standalone}
\usepackage{tkz-euclide}
\usetkzobj{all} 

\makeatletter
\pgfkeys{/@tkzautolab/.cd,
    center/.store in    = \tkz@center,
    with/.store in      = \tkz@with,
    with                = 0.15,
   /@tkzautolab/.search also   = {/tikz},
}
\def\tkzAutoLabelPoints{\pgfutil@ifnextchar[{\tkz@AutoLabelPoints}{%
                                         \tkz@AutoLabelPoints[]}}% 
\def\tkz@AutoLabelPoints[#1](#2){%
\begingroup 
\pgfqkeys{/@tkzautolab}{#1} 
 \foreach \point in {#2}{
 \path (\tkz@center) -- ($ (\point) + \tkz@with*($(\point)-(\tkz@center)$) $) node{$\point$};}
\endgroup
}%
\makeatother

\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(2,1){O} % Defines the point at which the origin is located
    \tkzGetRandPointOn[circle=center O radius 1.5cm]{A} % Gets random point of the circle with center at O and radius 1.5cm
    \tkzDrawCircle(O,A) % Draws the circle
    \tkzDefPointBy[rotation=center O angle 100](A) % Defines a point...
    \tkzGetPoint{C} % named C
    \tkzDefPointBy[rotation=center O angle 78](A) % Defines a point...
    \tkzGetPoint{B} % named B
    \tkzDrawPoints(O,A,B,C) % Draws dots
    \tkzDrawSegments(C,B B,A A,O O,C) % Draws the segments
    \tkzAutoLabelPoints[center=O,red](A,B,C)
\end{tikzpicture}
\end{document}

\documentclass[tikz,border=2pt]{standalone}
\usepackage{tkz-euclide}

\begin{document}
\begin{tikzpicture}
    \tkzDefPoint(2,1){O} % Defines the point at which the origin is located
    \tkzGetRandPointOn[circle=center O radius 1.5cm]{A} % Gets random point of the circle with center at O and radius 1.5cm
    \tkzDrawCircle(O,A) % Draws the circle
    \tkzDefPointBy[rotation=center O angle 100](A) % Defines a point...
    \tkzGetPoint{C} % named C
    \tkzDefPointBy[rotation=center O angle 78](A) % Defines a point...
    \tkzGetPoint{B} % named B
    \tkzDrawPoints(O,A,B,C) % Draws dots
    \tkzDrawSegments(C,B B,A A,O O,C) % Draws the segments
    \tkzDefBarycentricPoint(O=1,A=1,B=1,C=1)
     \tkzAutoLabelPoints[center=tkzPointResult,with=.3,red](O,A,B,C)
\end{tikzpicture}
\end{document}