Rewatching "The Name of the Doctor", and I still haven't figured out how Clara saves the Doctor. The Great Intelligence walks in to the Doctor's time stream and with his knowledge of the Doctor's life to turn all his victories in to defeats. Then Clara walks in after the GI and somehow her presence corrects everything. How? It's never explained and it doesn't make any sense. Did she just nullify the GI? If so how? Clara lost her memory when she was shattered in to millions of copies. The GI retained his.
Watching the two episodes that featured Clara before she became a companion proper; Asylum of the Daleks and The Snowmen. Clara's presence didn't thwart anything. Remove her from both episodes and there isn't much threat to the Doctor's life that she saves him from. The scenes with Clara edited in to clips of the Doctor's previous lives also don't affirm any contribution of her saving the Doctor.
Er, didn't Clara wipe all memory of the Doctor from the Dalek archives in Asylum? That sounds pretty useful, also she was
providing assistance for the Doctor, Rory and Amy that kept them alive.
As for the Snowman, she saves the Doctor by getting him involved, if she hadn't the GI's plan would have been more likely to succeed, and either the Doctor wouldn't have got inovled, or might have got involved later when there was less time to resolve the matter.
As for why her memory is splintered whilst the GI retains his. Firstly what proof is there that the GI retains his anyway, and secondly are you really debating how a creature that's pure thought, hundreds, potentially thousands of years old, composed entirely of information can do something a normal young human female in her 20s can't?
The Doctor doesn't have to hear her, doesn't need to even see her, all she needs to do is trip up the GI on his way to foil the Doctor's plans!