Мы так и не смогли придумать название для этого проекта.

Так и не is a very common Russian pattern that adds the idea of in the end still not or never did manage to.

So:

• Мы не смогли придумать название... = We couldn’t come up with a name...
• Мы так и не смогли придумать название... = We still never managed to come up with a name...

It often suggests that there was an expectation, some effort, or enough time for it to happen, but it never happened anyway.

These are related but not identical.

• не могли = were not able / couldn’t, often describing a general situation or circumstances
• не смогли = didn’t manage to, focusing on the result of a specific attempt

Here, не смогли придумать means they tried or intended to do it, but failed to achieve the result.

So Мы так и не смогли придумать название sounds like:

• we kept trying, or
• it was supposed to happen,
• but in the end it didn’t.

That is why смогли is very natural here.

Смогли is the past tense plural form of смочь.

Breakdown:

• dictionary form: смочь = to be able to, to manage to
• past masculine singular: смог
• past feminine singular: смогла
• past neuter singular: смогло
• past plural: смогли

Because the subject is мы (we), the verb must be plural, so смогли is used.

A useful thing to remember: in the Russian past tense, verbs agree with the subject in gender and number, not person.

Because смочь is like to manage to or to be able to, and it is normally followed by another verb in the infinitive:

• смочь сделать = to manage to do
• смочь найти = to manage to find
• смочь придумать = to manage to come up with

So:

• смогли tells you about ability/success
• придумать tells you what action they were trying to do

This is very similar to English managed to come up with.

Because придумать is perfective, and here the speaker is talking about reaching a completed result: coming up with a name.

• придумать = to come up with, to think of successfully
• придумывать = to be coming up with, to invent/think up in process, or repeatedly

Since the goal was a finished result — an actual name — perfective придумать fits best.

Compare:

• Мы не смогли придумать название.
  We failed to come up with a name.

• Мы весь вечер придумывали название.
  We spent the whole evening trying to think up a name.

The second sentence focuses on the process; the first focuses on the missing result.

Название is the direct object of придумать, so it is in the accusative case.

However, название is:

• singular
• neuter
• inanimate

For many neuter inanimate nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: название
• accusative: название

That is why it does not visibly change.

Because the preposition для requires the genitive case.

So:

• для = for
• проект → genitive singular проекта
• этот must agree with проект, so it becomes этого

That gives:

• для этого проекта = for this project

So the structure is:

• название = a name/title
• для этого проекта = for this project

Yes, that is also possible, but the nuance is slightly different.

• придумать название для этого проекта emphasizes finding a suitable name for the project
• придумать название этого проекта can also mean come up with this project’s name

In many contexts both are natural, but для этого проекта strongly highlights purpose: a name for the project.

So in this sentence, для этого проекта works very well because the idea is choosing a name to give to it.

Yes, often it can.

Russian frequently leaves out the subject when it is clear from context. So in conversation you might hear:

• Так и не смогли придумать название для этого проекта.

That can still mean We still couldn’t come up with a name for this project, if we is understood already.

Including мы can:

• make the sentence clearer,
• sound a bit more explicit,
• or add slight emphasis on we.

The word order is natural and neutral, but Russian word order is flexible.

This version:

• Мы так и не смогли придумать название для этого проекта.

sounds straightforward and normal.

Other orders are possible for emphasis, for example:

• Название для этого проекта мы так и не смогли придумать.

This puts more focus on the name for this project.

So the meaning stays basically the same, but the information focus can shift depending on word order.