PCs can speak languages. Generally, they are given Common and one other than fits their race (e.g. for a dwarf: "You can speak, read and write Common and Dwarvish", from PHB, pg. 20). There are a few races who are simply given a choice of any language, such as humans or tabaxi, and others who get a free langauge, such as half-elves or high elves. Furthermore, some backgrounds allow you to learn one or two extra languages.

For the cases where you get to pick a language, must you pick the whole language, or can you choose to know only one dialect of that language? In particular, I'm thinking of Primordial and its dialects Aquan, Auran, Ignan and Terran. From the PHB, pg. 123:

Some of these languages are actually families of languages with many dialects. For example, the Primordial language includes the Auran, Aquan, Ignan, and Terran dialects, one for each of the four elemental planes. Creatures that speak different dialects of the same language can communicate with one another.

If a PC has an in-character reason why they would only have learnt one dialect, such as a half-elf who is half-sea elf and was raised by the sea elves and thus learned only Aquan, or a human who spent some time among Azer or Salamanders or something and thus speaks Ignan, these characters would have no exposure to the other dialects, so picking Primordial wouldn't make sense.

There is some precedence for PCs only speaking certain dialects, since there are certain playable races, such as the Kenku (Volo's Guide to Monsters, pg. 111) and Aarakocra (Elemental Evil Player's Companion, pg. 5) who both speak Auran, not Primordial. On the other hand, Genasi (Elemental Evil Player's Companion, pg. 9) all speak Primordial, regardless of which subrace they are (even though it could be argued that it would make sense for them to only speak the dialect that matches their element).

When a player picks a language for their PC, can they choose just a single dialect of a language instead of the whole language, such as Primordial?

PCs can speak languages. Generally, they are given Common and one other than fits their race (e.g. for a dwarf: "You can speak, read and write Common and Dwarvish", from PHB, pg. 20). There are a few races who are simply given a choice of any language, such as humans or tabaxi, and others who get a free langauge, such as half-elves or high elves. Furthermore, some backgrounds allow you to learn one or two extra languages.

For the cases where you get to pick a language, must you pick the whole language, or can you choose to know only one dialect of that language? In particular, I'm thinking of Primordial and its dialects Aquan, Auran, Ignan and Terran. From the PHB, pg. 123:

Some of these languages are actually families of languages with many dialects. For example, the Primordial language includes the Auran, Aquan, Ignan, and Terran dialects, one for each of the four elemental planes. Creatures that speak different dialects of the same language can communicate with one another.

If a PC has an in-character reason why they would only have learnt one dialect, such as a half-elf who is half-sea elf and was raised by the sea elves and thus learned only Aquan, or a human who spent some time among Azer or Salamanders or something and thus speaks Ignan, these characters would have no exposure to the other dialects, so picking Primordial wouldn't make sense.

There is some precedence for PCs only speaking certain dialects, since there are certain playable races, such as the Kenku (Volo's Guide to Monsters, pg. 111) and Aarakocra (Elemental Evil Player's Companion, pg. 5) who both speak Auran, not Primordial; there's also the aforementioned sea elves (Mordenkainen's Tome of Foes, pg. 62) who speak Aquan. On the other hand, Genasi (Elemental Evil Player's Companion, pg. 9) all speak Primordial, regardless of which subrace they are (even though it could be argued that it would make sense for them to only speak the dialect that matches their element).

When a player picks a language for their PC, can they choose just a single dialect of a language instead of the whole language, such as Primordial?