What’s the proper soldering iron temperature for standard .031" 60/40 solder?
There is no proper soldering iron temperature just for a given type of solder - the iron temperature should be set for both the component and the solder.
When soldering surface mount components, a small tip and 600F (315C) should be sufficient to quickly solder the joint well without overheating the component.
When soldering through hole components, 700F (370C) is useful to pump more heat into the wire and plated hole to solder it quickly.
A negative capacitor lead to a heatsinking solid pour ground plane is going to need a big fat tip at a much higher temperature.
However, I don't micromanage my soldering temperature, and simply keep mine at 700F (370C). I'll change the tips according to what I'm soldering, and the tip size really ends up determining how much heat gets into the joint in a given period of contact.
I think you'll find that very few soldering jobs will really require you to change your tip temperature.
Keep in mind that the ideal situation is that the soldering iron heats up the joint enough that the joint melts the solder - not the iron. So the iron is expected to be hotter than the melting point of the solder so that the entire joint comes up to the melting point of the solder quickly.
The more quickly you bring the joint temperature up and solder it, the less time the soldering iron is on the joint, and thus the less heat gets transferred to the component. It's not a big deal for many passive or small components, but it turns out that overall a higher tip temperature results in faster soldering and less likely damage to the component being soldered.
So if you do use higher tip temperatures, don't leave them on components any longer than necessary. Apply the iron, apply the solder, and remove both - it should take just a second or maybe two for surface mount, and 1-3 seconds for a through hole part.
Please note that I'm talking about prototyping, hobbyist, and one-off projects. If you are planning on doing final assembly with the iron, repair work for critical projects, etc, then you'll need to consider what you're doing more carefully than this general rule of thumb.
As part of your problem (along with ground and power planes sucking heat away from the joint), your solder may well be lead based, and the solder on the graphics card is almost certainly lead-free (everything in the last few years has moved to lead-free solders). The lead-free solders require significantly higher temps to melt than older lead-based solders.
Unfortunately, I think the basic problem is that your iron is just not up to the task. The pre-heat trick might help, but I'd recommend just getting an iron that's got more punch.
Best Answer
If you look in the datasheet, you'll find a timed temperature profile (or a link to where that information is available on the manufacturer's web site) for machine soldering.
The temperatures will go above 175C, and for many seconds. The effects of this one time thermal abuse have been built into the specifications for the device, it has been designed to be machine soldered.
Unfortunately, hand soldering is much less controllable than a machine. However, several things are on your side. You can used leaded solder, which melts at a lower temperature than lead-free (the machine profile is certainly for lead-free). You can observe the tab, and stop heating moments after the solder flows.
As there's a lot of mass to heat, use the biggest soldering iron tip you can find. Use a controllable iron, and set the temperature to enough to melt the solder, but <= 260C. Practice on some dead devices. If you haven't any dead ones, practice on some live ones, and you'll soon get some dead ones. If you can do the job keeping the device within the time/temperature profile of the data sheet, then you're in with a good chance.
Where the device has an accessible tab, like a TO220, it's fairly easy. Where it's a power SO8, you can't heat the tab directly. With an etched board, the heat has to be applied from below, and the device reflowed on. When I don't have a board, I often 'dead-bug' the device, and solder heavy wires to the tab for the heatsink.