Мы так и не решили, оставить ли котёнку старую кличку или придумать новую.

Так и не adds emphasis to the negation. It means something like:

• we still didn’t
• we never did
• we ended up not

So Мы так и не решили is stronger than just Мы не решили.

• Мы не решили = we didn’t decide
• Мы так и не решили = we never actually managed to decide / we still hadn’t decided in the end

It often suggests that a decision was expected, attempted, or should have happened, but it did not.

Russian is focusing on the result: whether a decision was reached.

• решить (perfective) = to decide, to reach a decision
• решать (imperfective) = to be deciding, to work on deciding

So:

• Мы не решили = we did not reach a decision
• Мы не решали = we did not even try to decide / we were not deciding

In this sentence, the point is not the process, but the fact that no final decision was made. That is why решили is used.

Because everything after it is a subordinate clause: an indirect question.

The structure is basically:

• Мы так и не решили, ...
• We still didn’t decide, ...

And the subordinate clause is:

• оставить ли котёнку старую кличку или придумать новую

Russian normally uses a comma before this kind of indirect-question clause.

Ли is a particle used to introduce an indirect yes/no question, often translated as whether.

So:

• оставить ли котёнку старую кличку = whether to keep the kitten’s old name

In this sentence, the clause contains two alternatives:

• оставить ли ... или придумать ...
• whether to keep ... or come up with ...

So ли is helping express the idea of uncertainty or choice.

That is how ли normally works in Russian: it comes after the word it is attached to.

So Russian says:

• оставить ли not
• ли оставить

You can think of ли as a little clitic particle that usually goes in second position relative to the part being questioned or highlighted.

Here the questioned action is оставить, so ли follows it directly.

Because the kitten is not the direct object here; the name is.

The direct objects are:

• старую кличку
• новую (= новую кличку)

The kitten is the one for whom the name is kept or invented, so Russian uses the dative:

• оставить котёнку старую кличку = leave the kitten its old name
• придумать котёнку новую кличку = come up with a new name for the kitten

This is very natural in Russian with verbs connected to giving, assigning, or keeping something for someone.

They are in the accusative feminine singular because they are the direct objects of the infinitives:

• оставить what? → старую кличку
• придумать what? → новую (= новую кличку)

The noun кличка is feminine, so the adjective agrees with it:

• nominative: старая кличка
• accusative: старую кличку

And in the second half, Russian omits the repeated noun:

• придумать новую = придумать новую кличку

That omission is very normal.

Because it applies to both infinitives and does not need to be repeated.

The full underlying meaning is:

• оставить котёнку старую кличку или придумать котёнку новую кличку

But Russian, like English, often leaves out repeated words when they are clear from context:

• оставить ли котёнку старую кличку или придумать новую

This makes the sentence more natural and less heavy.

Кличка is a name or nickname for an animal. In this sentence it is the most natural word because we are talking about a kitten.

• кличка = an animal’s name
• имя = a personal name, usually for a human, though pet owners may sometimes also use имя for a pet

So старая кличка means the kitten’s existing or previous name.

One extra note: when used about a person, кличка often means nickname and can sound rough or unflattering.

Because they depend on решили in an indirect-question structure.

The idea is:

• We didn’t decide whether to keep... or invent...

Russian often uses infinitives after verbs like решить when talking about deciding what action to take.

So the pattern is:

• решить, что делать
• решить, ехать ли
• решить, оставить ли ... или придумать ...

In English, this often corresponds to whether to do X.

Literally, yes, оставить often means leave, but in this context it is best understood as:

• keep
• leave as it is

So:

• оставить котёнку старую кличку = keep the kitten’s old name

This is a very common Russian use of оставить: to leave something unchanged or keep something in its current state.