Why can't you consider coideal generated by sets. The therm "coideal generated by a set" don't exist in literature but didn't found anything explaining why, so i formulated an example of a 6-dimensional coalgebra in wich there's a 1-dimensional subspace $V$ that is not a coideal and is contained in two 2-dimensional coideals such that $V_1 \cap V_2 = V$. So the problem is that, given $X \subseteq ker(\epsilon)$, the nonempty family  $\{ I \supseteq X \colon I \text{ coideal of } C\}$ have minimal elements, but can have many of them. Development below:
 A: Let $S = \{a,b,c\}$ with order $a \prec b \prec c$ and let $C$ be the $k$-vector space with basis $B = \{ (x,y) \in S \times S \colon x \prec y \} $. Then $C$ have a coalgebra structure (the locally finite poset coalgebra) defined by $\Delta(x,y) = \sum (x,z) \otimes (z,y)$ where $x \prec z \prec y$ and $\epsilon(x,y) = \delta_{x,y}$. For simplicity i'll denote (x,y) = xy and use the sigma notation $\Delta(xy) = xz \otimes yz$. To verify that $C$ is a coalgebra we have
$$ (\Delta \otimes id)\Delta(xy) = (\Delta \otimes id)( xz \otimes zy ) = xw \otimes wz \otimes zy 
\hspace{25px}\text{with}\hspace{25px}  x \prec w \prec z \prec y $$
$$ (id \otimes \Delta)\Delta(xy) = (id \otimes \Delta)( xw \otimes wy ) = xw \otimes wz \otimes zy 
\hspace{25px}\text{with}\hspace{25px}  x \prec w \prec z \prec y $$
so $\Delta$ is coassociative, and
$$ (\epsilon \otimes id)\Delta(xy) = (\epsilon \otimes id)(xz \otimes zy) = \delta_{x,z} \otimes zy = xy $$
$$ (id \otimes \epsilon )\Delta(xy) = (id \otimes \epsilon )(xz \otimes zy) = xz \otimes \delta_{z,y} = xy $$
so the counity axiom holds. Now, consider $V$ the 1-dimensional vector space generated by $ac$. By definition we have
$$ \Delta(ac) = aa \otimes ac + ab \otimes bc + ac \otimes cc $$
where the first and last summands are in $C \otimes V$ and $V \otimes C$, but the middle one isn't in $C \otimes V + V \otimes C$, so $V$ can't be a coideal. Let $V_1$ be the subspace generated by $\{ac, ab\}$ and $V_2$ the subspace generated by $\{ac, bc\}$, so $V_1 \cap V_2 = V$ and
$$ \Delta(ac) \in C \otimes V + V_1 \otimes C + V \otimes C \subseteq V_1 \otimes C + C \otimes V_1 $$
$$ \Delta(ac) \in C \otimes V + C \otimes V_2 + V \otimes C \subseteq V_2 \otimes C + C \otimes V_2 $$
furthermore we have
$$ \Delta(ab) = aa \otimes ab + ab \otimes ab \in C \otimes V_1 + V_1 \otimes C $$
$$ \Delta(bc) = bb \otimes bc + bc \otimes cc \in C \otimes V_2 + V_2 \otimes C $$
so by linearity of $\Delta$ and the fact that $\epsilon(ac) = \epsilon(ab) = \epsilon(bc) = 0$ we got that $V_1$ and $V_2$ are both coideals of $C$.
