maybe you can overclock the core2 6600?
if your mobo allows it, that will help to somewhat extent
is it core 2 duo or core 2 quad? there is a c2d e6600 and a c2q q6600
guide for c2 duo
http://hexus.net/tech/tech-explained/cpu/9808-intel-core-2-duo-basic-overclocking-guide-beginners/
here is a guide for the quad
http://archive.atomicmpc.com.au/forums.asp?s=2&c=27&t=496
core 2 duo 2,4GHz is exactly whar they put as minimum requirement, if you have that, you might want to get a c2q quad used from ebay or some other market place
BF4 likes to have more than two threads, especially in multiplayer, oc c2q should be ok
should you have the c2d and it shows to be inadequate even with oc, this is about the cheapest possibility to go for something new with pretty good performance mobo+cpu
http://www.newegg.com/Product/ComboD...=Combo.1519915
or this
http://www.newegg.com/Product/ComboD...=Combo.1519941
the amd bundle will perform better in BF4 and crysis 3, the intel in other games, like starcraft 2 or anno 2070
this would be a real powerhouse intel quadcore + mobo, flawless performance, but that would crash the budget alone
http://www.newegg.com/Product/ComboD...=Combo.1519924
not sure, but you´d probably also need new ram, if you go for new mobo, like
http://www.newegg.com/Product/Produc...82E16820231740
and get the video card HD7870 or r9 270x, something like that, like they said
and if you have to get the cpu, maybe r9 270 without the x, it´s much closer to 270x than the next downgrade 260x
there are some out there in the 40-50$ range (auctions)
http://www.ebay.com/itm/Intel-Core-...sor-/191022966398?pt=CPUs&hash=item2c79db267e
edit: what kind of a psu (power supply) do you have?
it might be isufficient for anything more than a hd7750, which doesn´t need a pcie 6pin supply cable
(the r9 270x needs two, r7 270 and r7 260x needs one, r7 250 needs none)
there are adapters and they are usually included with the video card, but it still depend, whther the psu is strong/good enough
edit: as the gts 250 needs 1 pcie supply cable you can use a card that needs one, too:r9 270 (no x), r7 260x, hd7850 or gtx 650ti boost or even gtx 660
(hd 7790 is about the same as r7 260x, should you see a decisively cheaper one)
the r9 270 seems to be the fastes of the cards with need for one pcie cable, followed by gtx 660
(there are 1GB and 2GB versions of some of the cards, get the 2GB if it´s gddr5, don´t get any ddr3, even if that is more)
conclusion:
try oc c2d / c2q (maybe replace c2d, if you find c2q under ~50-60$)
if that´s not good, get a new mobo+cpu
get r9 270
or get
r9 270x/ hd 7870, if your psu has 2 pcie power supply cables, or at least one free molex for use with adapter cable, which is usually included with the r9 270/hd 7870
some amd cards come as a bundle with BF4 at a small charge for example
http://www.newegg.com/Product/Produc...82E16814150687
edit: i see,you´re already playing the game
anyway, the graphics card will be the bigger uprade, don´t get new mobo+cpu and no new card
get card first, then try or oc cpu then try
edit: just now realized you are gonna put the hd 6900 int your son´s pc, lol (wrote this for like two hours...next time more careful read first)
might still try the oc thing, if you like
for 200$, get the r9 270x or 7870, as the others suggested (edit: for your own pc, then put 6900 into son´s pc)