What is Roblox using for the Custom Chat Name Color Algorithm?

This is one of those very few questions I give a direct answer to, but I'm only doing this because just about every custom chat system on ROBLOX that I know of that implements this feature, uses the exact same method.

Basically, it's a combination between the length of the name, and each individual character's ASCII value. After it's gotten that information from within the for loop, well, the math below speaks for itself. Here's my version of it, where the name is the first parameter, and an array of Color3 values, is it's second:

local function GetLengthColor(String,ColorList)
    local Result = 0 -- Final number to modulate the #ColorList from
    for i = 1, #String do
        local ASCII = String:sub(i,i):byte() -- Get the ASCII value of it's current character
        local Comp = #str-i+1
        if #String%2 == 1 then
            Comp = Comp-1
        end
        if Comp%4 >= 2 then
            ASCII = -ASCII
        end
        Result = Result+ASCII
    end
    return ColorList[Result%#ColorList+1] -- Return the index value from the ColorList.
end

Hope this is what you were looking for. I'm sorry to say that I don't know what colors ROBLOX uses for their speaker tags, but this is the correct sequence in which it's chosen from.

If you have any questions regarding how the function works, just let me know and I'll be happy to explain.

To add onto ScriptGuider's answer:

These are the colors:

local CHAT_COLORS =
{
    Color3.new(253/255, 41/255, 67/255), -- BrickColor.new("Bright red").Color,
    Color3.new(1/255, 162/255, 255/255), -- BrickColor.new("Bright blue").Color,
    Color3.new(2/255, 184/255, 87/255), -- BrickColor.new("Earth green").Color,
    BrickColor.new("Bright violet").Color,
    BrickColor.new("Bright orange").Color,
    BrickColor.new("Bright yellow").Color,
    BrickColor.new("Light reddish violet").Color,
    BrickColor.new("Brick yellow").Color,
}

And this is the formula:

function getChatColor(pName)
    return CHAT_COLORS[(GetNameValue(pName) % #CHAT_COLORS) + 1]
end

Taken from the core script for the Roblox chat system.