When is an integrally closed local domain a valuation ring? Every valuation ring is an integrally closed local domain, and the integral closure of a local ring is the intersection of all valuation rings containing it. It would be useful for me to know when integrally closed local domains are valuation rings.
To be more specific,

is there a property $P$ of unitary commutative rings that is strictly weaker than being a valuation ring, such that an integrally closed local domain is a valuation ring iff it satisfies the property $P$.

 A: A commutative ring $R$ is called coherent, if every finitely generated ideal $I$ is finitely presented, that is as an $R$-module $I$ is isomorphic to $R^n/J$ for some finitely generated $R$-submodule $J$ of $R^n$.
For two ideals $I,J$ of $R$ one defines the ideal $(I:J):=\{r\in R : rJ\subseteq I\}$.
Now the following is true: the local integrally closed domain $R$ is a valuation domain if and only if $R$ is coherent and there exist $r,s\in R$, $s\not\in rR$ such that the maximal ideal $M$ of $R$ is minimal among the prime ideals containing $(rR:sR)$.
This follows from results obtained by J. Mott and M. Zafrullah some decades ago.
References:
S. Glaz, Commutative coherent rings, Lecture notes in mathematics 1371, 1989.
(general theory of coherence)
J. Mott, M. Zafrullah, On Prüfer -v-multiplication domains, Manuscripta Mathematica 35 (1981). (Theorem 3.2 is relevant)
M. Zafrullah, On finite conductor domains, Manuscripta Mathematica 24 (1978). (Theorem 2 is relevant)
A: Note the following statements.
I. A quasi local domain $(D,M)$ is a valuation domain if and only if $D$ is
a Bezout domain (i.e. for every pair $a,b$ in $D,$ the ideal $(a,b)$ is
principal or, equivalently, every finitely generated ideal of $D$ is
principal).
If $D$ is a valuation domain then as for each pair $a,b$ in $D$ we have $a|b$
or $b|a$, giving $(a,b)=(a)$ or $(a,b)=(b)$, that is $D$ is Bezout.
Conversely, take $a,b$ in $D.$ If either of $a,b$ is zero or a unit $a|b$ or 
$b|a.$ So, let both $a,b$ be nonzero non units. Since $D$ is Bezout, $%
(a,b)=(c)$ for some $c$ in $D.$ Clearly $c|a,b.$ Let $a=a^{\prime }c$ and $%
b=b^{\prime }c.$ Substituting, we get $(a^{\prime }c,b^{\prime }c)=(c)$.
Canceling $c$ from both sides we get $(a^{\prime },b^{\prime })=D.$ As in a
quasi local domain nonzero non units generate a proper ideal, at least one
of $a^{\prime },b^{\prime }$ is a unit. So, $a^{\prime }|b^{\prime }$ or $%
b^{\prime }|a^{\prime }$ leading to $a^{\prime }c|b^{\prime }c$ or $%
b^{\prime }c|a^{\prime }c$ and to $a|b$ or $b|a.$
II. A quasi local domain $(D,M)$ is a valuation domain if and only if $D$ is
a Prufer domain (every two generated nonzero ideal is invertible or ,
equivalently, every finitely generated nonzero ideal is invertible).
Follows from I. once we note that in a quasi local domain each invertible
ideal is principal.
Note that P: $D$ is Bezout or P: $D$ is Prufer both are non-trivial in that
there are Bezout (resp., Prufer) domains that are not valuation domains. So
perhaps that would suffice as an answer.
Now the above two results do not require the domain $(D,M)$ to be integrally
closed and you are asking for a property P such that $(D,M)$ is a valuation
domain. Here is the exact property P: Every finitely generated nonzero ideal
of $D$ is a $v$-ideal (i.e. a divisorial ideal). So we have the statement.
III. An integrally closed quasi local domain $(D,M)$ is a valuation domain
if and only if every nonzero finitely generated ideal of $D$ is a $v$-ideal.
For the proof look up Theorem 8, on pages 1710-1711, of an old paper of
mine: [Z] The $v$-operation and intersections of quotient rings of integral
domains, Comm. Algebra, 13 (8) (1985) 1699-1712.
The cited theorem says: An integrally closed fgv domain is a Prufer domain. 
Now fgv domain is a fancy name for a domain whose nonzero finitely generated
ideals are divisorial. Indeed as every invertible ideal is divisorial the
converse of Theorem 8 of [Z] is also true. You can also get information on
divisorial ideals (i.e. $v$-ideals) from [Z] or sources mentioned there.
Proof of III. Let $(D,M)$ be an integrally closed domain such that every
nonzero finitely generated ideal of $D$ is divisorial. Then $D$ is a Prufer
domain by Theorem 8 of [Z] and by II. above $D$ is a valuation domain.
Conversely let $(D,M)$ be a valuation domain then every nonzero finitely
generated ideal of $D$ is principal and so divisorial.
Note:If you would rather follow Hagen' suggestion here's how to go about it. Note that a nonzero ideal $A$ is a $t$-ideal if for each finitely generated nonzero ideal $I$ contained in $A$ the $v$-image (I sub v) is also contained in $A$.
So Hagen wants you to use P: $(D,M)$ is such that $D$ satisfies FC and $M$ is a $t$-ideal.
An easy way to see what Hagen means is look up Lemma 5 of the finite conductor domains paper mentioned above by him. 
A: A very late answer, but I think there are some other connections and references that merit mention and maybe will interest someone down the line.
As mentioned by M. Zafrullah, in the presence of the local condition we can reduce the search to a property making integrally closed domains Prüfer.
The first connection I'll point out was explained by Zafrullah in his post, but here is another way of thinking about it.

Let $D$ be a domain. For a polynomial $f \in D[x]$, let $c(f)$ denote the ideal generated by its coefficients$^1$.
$(1)$ $D$ is Prüfer iff all polynomials $f,g \in D[x]$ satisfy $c(f)c(g) = c(fg)$
$(2)$ $D$ is integrally closed iff all polynomials $f,g \in D[x]$ satisfy $[c(f)c(g)]_v = c(fg)_v$

A simple property that bridges (1) and (2) is the requirement that every finitely generated ideal is divisorial, so that's one possible answer to the question.
Another approach is to just jack all the results about when the integral closure of a domain is Prüfer, and there are many nice characterizations of that. First to lay out a couple of definitions.... A prime $P$ of $D[x]$ such that $P \cap D = 0$ is called an upper to zero. A domain $D$ is called a UMt if every upper to zero contains a polynomial $f$ such that $c(f)_v = D$, or equivalently if every upper to zero is a maximal $t$-ideal (the terminology UMt comes from Uppers to zero are Maximal t-ideals).  Now here is a sample of results on Prüfer integral closure.

Let $D$ be a domain with field of fractions $K$, and let $D'$ be the integral closure.  The following are equivalent$^2$.
$(a)$ $D'$ is Prüfer.
$(b)$ $D'$ is a QQR-domain, i.e. every overring of $D'$ is the intersection of localizations of $D'$.
$(c)$ Every overring of $D$ is a UMt.
$(c')$ $D$ is a UMt and the maximal ideals of $D$ are $t$-ideals.
$(c'')$ Every upper to zero contains a polynomial $f$ having $c(f) = D$.
$(d)$ Every $k \in K$ is the root of a polynomial $f \in D[x]$ having $c(f) = D$
$(e)$ For every $k \in K$, the extension $D \subseteq D[k]$ satisfies INC, i.e. if $P,Q$ are primes of $D[k]$ such that $P \cap D = Q \cap D$, then $P,Q$ are incomparable.

These give some nice non-trivial answers to the question of a when a local integrally closed domain is a valuation ring.  For example, (d) says that a local integrally integrally domain is a valuation ring iff every element of the fraction field is the root of some polynomial having a unit coefficient
and $(b)$ enables us to check if the domain is a valuation ring by checking to see if there are 'complicated' overrings that can't be achieved by intersecting localizations.
Another approach we could take is related to the last one. Note that valuation rings have the property that every ideal is a $t$-ideal.  So we might as well check that the maximal ideal of our local ring is a $t$-ideal right off the bat. If not, then we don't have a valuation ring at hand.  But if yes, then it suffices to show that our ring is a PvMD, more generally than a Prüfer domain, and the analog of the above results for PvMDs then becomes available.  For example, a local integrally closed domain is a valuation ring iff its maximal ideal is a $t$-ideal and every element of the fraction field is the root of a polynomial $f$ having $c(f)_v = D$, etc. etc.
$^1$ A standard reference is Gilmer's Multiplicative Ideal Theory.  (1) is covered in 28.1-28.6, and (2) is 34.8.  The main ingredients in these characterizations are the Dedekind-Mertens content formula for polynomials and the fact that the integral closure is the intersection of valuation overrings.
$^2$ $(a) \iff (b)$  Easy using the characterization of Prüfer domains as having every overring integrally closed, and again using that integrally closed domains are the intersection of their valuation overrings (which for Prüfer rings are localizations at primes).  $(a) \iff (c,c',c'')$ See here for the originating work and also section $1$ of this paper which also mentions most of the results in this post.  Note especially in $c'$ how the assumption that maximal ideals are $t$-ideals implies that $I_t = D \implies I = D$, which interacts with the UMt property by forcing every upper to zero to contain a polynomial with a unit coefficient.  This has a similar effect to our first approach of requiring f.g. ideals to be divisorial.  $(a) \iff (d)$ See Theorem 5 here $(d) \iff (e)$ Actually this could be stated more strongly for individual elements, which are usually called primitive, and for arbitrary ring extensions.  See the first theorem here for details.
