UGH! Do not stick a pin in it unless you have boiled it for 10 minutes or so - that will give you a wicked infection!
Having said that - I really have no idea what that would be, given the location. You need to get it looked at. Bartholin glands are located on either side of the vaginal opening.
What colour is the lump? Is it red or inflamed or flesh coloured? I think you will need to have it looked at.
Bartholins Glands are on the inside sides of the vagina.
She sounds like she's got a boil going on for sure if she can stick a pin in and have pus come out.
Here is some information on boils and how to treat them:
http://www.healthy-skin-guide.com/boil-treatment.html
If its painful and really swollen they have topical antibiotics to apply on them.
Ouch!
When you poke it what color is the fluid? If it's clear then it isn't a boil or a pimple. Most likely wouldn't be ingrown hair if it's clear either. Might be a cyst. STOP poking it. Warm compress it for a few days and see if it goes down, if it still hurts or it hurts too much get it checked out.
It may be a pimple. But also, google bartholins cyst. I am not sure about the spelling. Try a warm compress, instead of poking at it. You don't want it to become infected.
Could it be a large pimple or boil? That's what it sounds like to me. Or even an infected ingrown hair.