Is every accessible category well-powered? Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.
Every accessible category with pushouts (hence also every locally presentable category) is well-copowered: this is shown in Theorem 2.49 of Locally presentable and accessible categories by Adámek and Rosický, and in Proposition 6.1.3 of Accessible categories by Makkai and Paré.  The question of whether all accessible categories are well-copowered seems to depend on set theory (it follows from Vopenka's principle by Corollary 6.8 of Adámek and Rosický, and implies the existence of arbitrarily large measurable cardinals by Example A.19).
Are all accessible categories well-powered?  I have been unable to find a mention of this one way or the other in either of these standard references.
 A: It seems to me that every category with a small set of dense generator is well powered. In particular accessible categories are well powered.
Dense generator means that you have a small full subcategory $C \subset A$ such that the induced nerve functor $A \rightarrow \widehat{C}$ is fully faithful. This functor send monomorphisms to monomorphisms so, as it is fuly faithful, isomorphisms class of subobjects in $A$ are a subset of the isomorphisms class of subobjects in $\widehat{C}$ (which is well powered). 
Note: this does not conflict with Ivan's answer, Strong generator only means the nerve is faithful and conservative and this is not sufficient to deduce that the Nerve is injective on iso class of mono. One can have two mono $A \subset X$ and $B \subset X$ with isomorphic image under the nerve functor, if none of them is included in the other then you can't deduce that $A$ is isomorphic to $B$.
Of course if one add the assumtpion that intersection of subobjects exists then one can use their intersection to compare them using only conservativity. i.e. if we have intersection of subobjects, a set of strong generator is enough, as mentioned by Ivan.
A: Some considerations, not a full answer (yet).


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*In Accessible categories and models for linear logic, at page 2, Barr claims that every accessible category is well powered. He even claims that is observed in the classical reference by Makkai-Parè. I did not manage to find it.

*A locally small category with finite intersections of subobjects and a (strong) generating set is well-powered, this appears in Johnstone [Sketches of an elephant, Remark A1.4.17]. Thus when an accessible category has finite intersections, it is well powered.
I still believe that every accessible category is well powered, and I am looking through the literature.


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*Observe that the statement that appears in Rosicky-Adamek on page 2, namely every locally small category with a strong generator is well-powered is wrong, as proved by the following counterexample.

