Learn Hardcore Turkish: Patron teslim tarihini değiştirebileceğini söyledi; ama son tarih aynen duruyor. - The boss said she might change the delivery date; but the deadline remains the same.

By default it’s the same as the subject of the main clause, so it refers back to Patron (the boss). In other words: “The boss said that he (the boss) could change it.” If you want the embedded clause to have a different subject, you mark that subject in the genitive:

Patron, asistanının teslim tarihini değiştirebileceğini söyledi. = The boss said that his assistant could change the delivery date. You can also emphasize that it was the boss himself with kendisinin:
Patron, teslim tarihini kendisinin değiştirebileceğini söyledi.

Why is değiştirebileceğini used instead of değiştirebilir after söyledi?

With söylemek, indirect/reported speech is typically formed with a nominalized clause: -(y)ecek/-(y)diğini + possessive + case. So you say:

Patron teslim tarihini değiştirebileceğini söyledi. If you want to use a finite verb like değiştirebilir, then switch to a “that”-style complement:
Patron dedi ki teslim tarihini değiştirebiliriz. Or use direct speech:
Patron: Teslim tarihini değiştirebilirim, dedi. Using … değiştirebilir dedi without ki/diye or quotation is not standard.

How is değiştirebileceğini built morphologically?

değiştir- = to change (transitive)
-ebil- = can/be able to (ability/possibility)
-ecek = future/irrealis participle (forms “would be able to change” in reported speech)
-(s)i = 3rd person possessive (turns the clause into a noun phrase: “his/her being-able-to change”)
-(n)i = accusative (because this whole clause is the object of söyledi) Note the buffer -n- appears between the 3rd person possessive and the accusative: …-i + -i → …-ini → değiştirebileceğini.

Why is teslim tarihini in the accusative?

Inside the embedded clause, değiştirmek is a transitive verb that takes teslim tarihi as its direct object. Therefore it’s marked accusative: teslim tarihini. The entire embedded clause (not just the object) is then the object of söyledi. By contrast, in the second clause son tarih is the subject of duruyor, so it’s unmarked.

What’s the difference between teslim tarihi and son tarih?

teslim tarihi = delivery date (the date something will be handed over/arrive).
son tarih = deadline (the latest permissible date). In many contexts you also see son teslim tarihi (“submission deadline”), which combines the ideas.

What exactly does aynen duruyor mean, and is it idiomatic?

It means “it remains exactly as it is/unchanged.” aynen is an adverb (“exactly/as is”), and duruyor often means “keeps standing/remains (unchanged)” in such contexts. It’s idiomatic and quite common, especially in speech. Neutral alternatives:

Son tarih değişmedi.
Son tarih aynı kaldı.

Could I say son tarih aynı duruyor?

That sounds off. aynı is an adjective; here you want an adverb. Use aynen duruyor, or switch the verb:

Son tarih aynı kalıyor.
Son tarih değişmedi.

What nuance does -ebil- add here? How would it differ from değiştireceğini söyledi?

-ebil- expresses ability/permission/possibility (“could”). So değiştirebileceğini söyledi = “said (he) could change (it).” Without -ebil-, değiştireceğini söyledi means “said (he) would change (it)”—a statement of intention/plan rather than ability/permission.

How do I say “The boss said it could be changed” (passive)?

Use the passive inside the embedded clause and make the embedded subject genitive:

Patron, teslim tarihinin değiştirilebileceğini söyledi. Here, teslim tarihinin is the genitive-marked subject of the embedded clause, and değiştirilebileceğini is the passive (“could be changed”).

Why are the tenses different: söyledi (past) but duruyor (present progressive)?

They refer to different time frames. The saying happened in the past (söyledi), while the deadline’s state is true now (duruyor = “is still remaining”). If you were talking about the situation at that past time, you could say duruyordu; if it’s about the future, duracak.

Is the semicolon before ama normal?

It’s acceptable but uncommon. Most writers would use a comma or a period:

… söyledi, ama …
… söyledi. Ama … In more formal writing, ancak or fakat can replace ama, and a semicolon can work before them:
… söyledi; ancak …

Can I replace söyledi with dedi here?

Not with this structure. demek prefers direct quotes or a ki/diye complement:

Direct quote: Patron: Teslim tarihini değiştirebilirim, dedi.
With ki: Patron dedi ki teslim tarihini değiştirebiliriz.
Indirect nominalized (your sentence type) works best with söyledi: … değiştirebileceğini söyledi.

Does Patron mean “my boss” here?

Literally it’s “the boss.” In context, if both speakers share the same workplace, Patron can pragmatically mean “our boss.” To be explicit:

patronum = my boss
patronumuz = our boss

Is teslim tarihi the only way to say this? What about teslimat tarihi?

Both occur. teslim tarihi is very common and general. teslimat tarihi is frequent in logistics/shipping. For deadlines relating to submissions, son teslim tarihi is the standard phrase.