Which developer? The effects of dilution depend on the
developer.
Deveoping time does not vary linearly with dilution. For
instance, diluting a developer to half strength results in
an increase in time for the same contrast of about 1.5, not
twice as one would expect. For developers which have a large
amount of sulfite this can result in an increase in grain
because some of the solvent action of the sulfite is lost.
For developers with low sulfite and relatively poor
buffering like Rodinal diluting will result in somewhat
lower grain due to the lower pH.
 At some point diluted developers will start to produce
border effects. This is the exageration of contrast at high
gradient areas of the image, that is, where there is a sharp
transition from low to high density. This is sometimes
attributed to "local exhaustion" but that is not really what
happens. As developers work they produce reaction products.
These can either accelerate or restrain the development
process depending on what the developing agents are and the
amount of sulfite. In normal development the reaction
products are limited by diffusion away from the area adn by
diffusion of new developer to the area. Where the developer
is sufficiently diluted there is not enough sulfite or other
preservative components to limit the action so there is a
tendency for the reaction products from the high density
area to diffuse to the lower density area and interfere with
development. Developers like D-76 will produced border
effects, also called adjacency effects, when diluted 1:3.
 Developers like Microdol-X and Kodak D-25, which depend on
strong solvent action for extra-fine-grain results lose
their extra fine grain properties when diluted. They also
gain some film speed since the loss of speed is due to the
solvent action destroying some of the latent image. At 1:3
these developers deliver grain and film speed about equal to
D-76 full strength.
 For lesser dilution there may not be much effect. D-76 at
1:1 has about the same grain and speed as full strength and
doesn't have significant border effects.
 Very highly diluted developers, for instance Rodinal at
1:200, can have compensating effects, that is, they stop
development of high density areas causing a shoulder (or
reduction of contrast) of the denser parts of the image.
This is sometimes desirable where extrememly high conrast
subjects have been photographed. This is again due to a
build up of reaction products in the emulsion. The idea of
stagnant or stand developing is similar. If the film is not
agitated reaction products can move from the area of
development only by diffusion through the developing
solution. That is very slow compared to the rate of
diffusion going throught only the emulsion as happens where
normal agitation is applied. Stand development results in a
somewhat distorted tonal renditon. It can also result in
very uneven development if the film is vertical in the
solution or there is any disturbance during development.
When vertical there is localized movement due to convection.
This can result in local directional streaking from denser
reaction products. Eventually, they diffuse into the general
body of the developer but remain in a cloud around the
surface of the film for long enough to cause problems.
 In general reasonable dilution can result in more
convenient development times and better uniformity without
having much effect on other characteristics such as grain.

That's an involved question. Depends on the developer, film, temp, agitation....

If you really want to know, I suggest you obtain a copy of Anchell and Troop's "The Film
Developing Cookbook." They cover this topic in pretty good detail.