The items that Syme asks Winston for are razor blades. There is a current shortage of them, as often happens with many necessities. This emphasizes the fact that life is not particularly comfortable for the ordinary Party members, in spite of the Party's continual assurances that there is great wealth and abundance for everyone:
The fabulous statistics continued to pour out of the telescreen. As compared with last year there was more food, more clothes, more houses, more furniture, more cooking-pots, more fuel, more ships, more helicopters, more books, more babies -- more of everything except disease, crime, and insanity. Year by year and minute by minute, everybody and everything was whizzing rapidly upwards (Part 1, chapter 5)
Such proclamations fly in the face of the evidence, as everything is actually drab and dreary, strictly regimented and rationed (particularly luxuries like chocolate); and yet, as Winston observes with dismay, most people are quite ready to accept whatever lies the Party feeds them.
WInston's reaction to Syme asking for razor blades also reveals something about his character. He actually has been hoarding some for himself but pretends to Syme that he doesn't have any extra ones at all. He appears to be on relatively friendly terms with Syme but he will not share anything with him. This illustrates how lonely and suspicious being a Party member can be. Winston is not able to confide in anyone except Julia, when she becomes his lover, and O'Brien - but O'Brien turns out to be the enemy.
Finally, we may note another significance to razor blades in this novel. When imprisoned in the Ministry of Love Winston longs for a razor blade in order to commit suicide, but even this is denied him. This everyday object could have become a means of escape for Winston, but it is not to be. The Party wields absolute control not just over its members' lives, but also over their deaths.