Are you asking what the patronymic or demotic might have been? Because none of those examples is a surname, and none of them works like a surname. It's not a part of Socrates' name that his father was Sophroniscus, that he was born in the deme of Alopece, that he belonged to the tribe Antiochis, or that he was an Athenian. Unlike a surname or a patronymic-as-surname none of these identifiers would have served as "the" way to identify Socrates. Any, all, or none of them could be used, and depending on the context and the speaker some might be more relevant than others. A non-Athenian speaking of Socrates would have just called him "the Athenian," whereas according to Diogenes Laertius Socrates' indictment by Meletus named him both by his patronymic and his demotic, as the son of Sophroniscus of the deme Alopece. These aren't surnames, in that none of them are part of Socrates' name, formal or otherwise.

If you're asking about the Greek forms used there's not much that can be said that will be terribly helpful unless you already know Greek and can produce the forms yourself. In which case you already knew this. It's typically thought that the earliest patronymic forms were hybrid adjectives ending in -ίδᾱς (Attic -ίδης) but even as early as Homer that ending is not always used as a patronymic in the strict sense and by the actual historical period it's just an adjective or noun type used most commonly in standalone names in their own right, with no indication of any kind of kinship (and not frequently appended to words that would stand as names on their own). Patronymics were indicated by the genitive of the father's name, which takes different forms depending on declension. In Socrates' case that would be "Σωφρονίσκου," lit. "of Sophroniscus." Somewhat less commonly you might find the patronymic used with the noun υἱός, "son," which was understood to be implied in any case. This might or might not include the article (ὁ Σωφρονίσκου υἱός vs. υἱός Σωφρονίσκου) and because Greek is an inflected language you might find the attribution being written differently (e.g. ὁ Σωκράτης ὁ Σωφρονίσκου υἱός or ὁ Σωκράτης ὁ υἱός Σωφρονίσκου). And then that form of course, which already changes according to declension, also changes according to case because again Greek is an inflected language. "Of Athens," or more accurately in Greek, "the Athenian," is pretty easy. You just use the adjectival form of the person's place of residence or birth. You could use either--some texts of Herodotus call him Herodotus of Halicarnassus whereas Aristotle referred to him as Herodotus of Thurium. These adjectives also differ in declension, and will change form according to case. For Socrates it's simply "ὁ Ἀθηναῖος." The demotic, which was what was typically used by Athenians and in official Athenian documents, is quite tricky. One might use an adjectival form of the deme name or one might just as typically use an adverb. These forms can be very funky and they can change. They show up quite often in epigraphy, so you either need to know the names of all the demes off the top of your head or you need your references on hand. Diogenes Laertius quotes the indictment against Socrates as using the adverb of his deme (Ἀλωπεκῆθεν) but in the same sentence Meletus' demotic is given as an adjective (Πιτθεὺς). It depends on a lot of things, and for Alopece in particular I've never encountered an inscription where it isn't an adverb, even though most other demotics are adjectives more commonly. Tribal names at Athens as elsewhere were usually done in the genitive, although sometimes you find inscriptions where the adjectival is used. Socrates' tribe would have been Antiochis since he was from the deme Alopece, and he says he's from Antiochis at Apol. 32b. Tribal names were not commonly used by Athenians except in certain circumstances (e.g. the prytany or military lists) although they were used frequently in other cities that didn't have demes. Like the demotics or the city adjectives you need to be able to construct either the adjective of the tribe or you need to know how to decline the tribal name, since either you used the adjective (not very common, at least in Attic inscriptions) or you stuck it in the genitive. The genitive singular of Antiochis is Ἀντιοχίδος and I have no idea what the adjective would be