How to remove remnants of a capacitor pin?
A large amount of copper is probably acting like a large heat-sink to the pin.
Getting enough heat into board to make joint properly molten all the way through is needed. While also minimising board overheating.

Resolder with new solder so there is a reasonable amount of solder present. More rather than less can help. 


Use a solder sucker. Heat solder until well molten and then apply sucker. Apply rapidly after iron tip is removed to ensure solder is as hot as possible. If you have somebody to help of if you have 3 or 4 arms try having sucker held ready by one person while the other heats the contact. Person A does a classic 3 2 1 go! and person  applies sucker and sucks. Avoid blocking hole with iron while sucking. More vertical than this in this case.



Professional desolderers that heat and then apply vacuum and vibration are also available. Thy work superbly compared to most other means. Finding someone local with one of these would save much playing and possible damage. Here is a cheap one:


Similar with solder braid. Lots of new solder will help the whole lot suck out by capillary action. If you have not used solder braid before try it out somwhere else first. There is a certain knack to using it well.
Impact. This can work very well.have board held in such a way that it can be swung against a flat surface so that there is significant impact shock.  Heat till molten - Impact shock - check/repeat. Bang board so as to maximise shock but present unintended stress at impact due to eg large force onto a small area. Don't hit so hard as to risk damaging the board.

Done properly this method is very good. Make that "extremely good". I have used it often over decades.The trick is to not beat the board to death but to give it a firm "slap". Banging a whole motherboard on a flat surface achieves this easily enough. Solder is high density and is expelled quite well by impact, tending to carry lead ends with it. Adequate heating is needed in this and all other cases. 

Method:

you probably won't find this method in most textbooks :-).


Find surface on which you can hit MB on the flat under surface so that it stops abruptly. Table top etc. 
Work out how you can hold MB so that it can be swung easily to impart "slap". 
Holding in an electrostatically safe manner is a very major bonus - well touching some grounded parts is probably enough. You wearing a grounding strap is extra points.  
A thick layer of newspaper would help protect table top and motherboard. 
Work out how hard you are happy to slap your MB on table top. 
Heat solder till molten.
Slap.
Repeat as required. 

Insert small drill down hole while heating. This can work well. Determine what size drill will go though hole comfortably. Probably 0.6mm to 0.8mm OK. maybe not. Small drills get hard to find and expensive.  Do not use a drill so large that it damages PTH barrel in through-holes. Hold in micro chuck such as eg older Dremels have.
Heat joint and push and twist with drill (by hand)  as required. Lead may push out easily or fight. Turning drill in hot solder will remove solder or lead progressively. Power drilling also works but should not be needed and is risking more damage 
Low temperature fractureable solder. Solder especially made for mechanical removal is available. This is usually used for IC renoval but may well work well enough here. Erode solder with dental pick or similar. One brand is Chip Quick.