Civilization – n advantage to building a road to a city-state before they request one

UPDATE The formula has changed as of the March 2011 patch, it is now

(city population * 1.1) + (capital population * .15) - 1 for each city connected to the capital, not including the capital.

• The Machu Pichu wonder increases the modifier by 20%, to (city population * 1.3) + (capital population * .15) - 1.
• Arabia increases the constant by 1, to (city population * 1.1) + (capital population * .15) (thanks WillfulWizard).

The economic overview also gives a far better explanation of how it is calculated - in other words, this question is now trivially solved by simply looking at that economic overview :)

Original answer below.

Okay, I did a bit more testing, and bwarner's answer is almost accurate:

Each city, excluding the capital, provides (city's population * 1.25) + 0.01 gold per turn. Owning the Machu Pichu wonder increases the modifier by 20%, to (population * 1.5) + 0.01, for all the cities.

The capital does not provide any gold.

The 1.25 and 0.01 can be seen in the Assets\Gameplay\XML\GlobalDefines.xml file:

<Row Name="TRADE_ROUTE_BASE_GOLD">
    <Value>1</Value>
</Row>
<Row Name="TRADE_ROUTE_CAPITAL_POP_GOLD_MULTIPLIER">
    <Value>0</Value>
</Row>
<Row Name="TRADE_ROUTE_CITY_POP_GOLD_MULTIPLIER">
    <Value>125</Value>
</Row>

The economic overview also demonstrates these values.

I couldn't find any other factor which affects these values, and I checked different difficulty levels, different distances between cities, different city route type (road vs railroad, road/railroad vs harbor), different city health and whether it is occupied. Looks like it's solely the population.

The diplomacy summary that you get from clicking the scroll in the upper right will show you all the civs and city states at once, as well as what resources they have available, how much gold they have, and how much gold per turn they are making.