Another approach:

In[1]:= theList = {{x1, y1}, {x2, y2}, {x3, y3}, {x4, y4}, {x3, y17}};

In[2]:= f[alist_, y_] := Cases[alist, {_, y}]

In[3]:= f[theList, y17]

Out[3]= {{x3, y17}}

Not that it is of any importance, but I just saw/realized that Lucas asked for a function that returned the second element, given the first. And I mistook it for the other way around. So this is just to set my record straight :)

In[1]:= theList = {{x1, y1}, {x2, y2}, {x3, y3}, {x4, y4}, {x3, y17}};

In[2]:= f[alist_, x_] := Cases[alist, {x, _}]

In[3]:= f[theList, x3]

Out[3]= {{x3, y3}, {x3, y17}}