Synesthesia for the rest of us (i.e. algorithm for unique string colorization)
Code Style


Where does the 32 come from? It seems strange to have AlphabetLength assigned as a constant, but 32 is an unexplained magic number. Especially if its value is actually based on the length of the alphabet, you should make it clear what the significance of that number is.
The method signature is pretty weird. Looking it up, apparently it's from some IValueConverter interface. You'd be better off extracting everything you have here into a private method which takes a string parameter and returns a SolidColorBrush. Then you can call that from Convert, doing any casting or validation you need there.
You have a mix of var and explicit type names. It'd be more consistent to declare your floats as var r = 0f, etc.
Your RGB values seem to be based entirely on the first, second and last characters of the string. That means there's really no reason to loop over it, you can just assign them directly
index is casted to a float everywhere it's used. It'd be cleaner to have it assigned as a float in the first place.


Algorithm


Currently, your names can be any colour at all. That's going to make displaying them hard, because you'll potentially have to modify the background colour to make sure the name reads clearly. You might be better off instead using HSV (Hue, Saturation, Value) colour space with a fixed V, then play with the H and S. It might also be good to use a restricted range of S so you don't end up with lots of washed-out, similar-looking colours
Using only the first, second and last characters in a name means you're going to have a lot of clashes, where different names show as the same colour. A different approach might be to do something like name.GetHashCode() to get an integer representing the string. You could then use that to generate the HSV (or RGB) components a few ways, like passing it as a seed to new Random (which will be consistent each time). Colour space is relatively limited, so you'll never get away from clashes altogether, but I think this will make them far less common.