عندي فكرة ثانية عن هذا المشروع.

Literally, عندي means at me or with me.

Arabic often expresses possession with a prepositional phrase instead of a verb like to have.

• عند = at / with
• ي = my / me

So:

• عندي = at me = I have

This is a very common Arabic pattern:

• عندي كتاب = I have a book
• عندها سيارة = She has a car

Because the ـي at the end of عندي already tells you the speaker is me / my.

Arabic often leaves out the independent pronoun أنا when it is not needed.

So:

• عندي فكرة ثانية = I have another/second idea
• أنا عندي فكرة ثانية = I have another/second idea

Both are possible, but adding أنا gives extra emphasis, like I have another idea.

This is a normal and very common Arabic sentence pattern for possession:

• عندي = I have
• فكرة = an idea

So the structure is basically:

• possessor phrase
  • thing possessed

Arabic is very comfortable beginning a sentence with a phrase like عندي. It sounds natural.

You could also say:

• لديّ فكرة ثانية عن هذا المشروع

That is a bit more formal in style, but the meaning is very similar.

Because in Arabic, adjectives usually come after the noun they describe.

So:

• فكرة ثانية = a second / another idea

Not:

• ثانية فكرة

Also, the adjective must match the noun. Here:

• فكرة is feminine singular
• so the adjective must also be feminine singular: ثانية

This matching is very important in Arabic.

Because فكرة is a feminine noun.

Many Arabic nouns ending in ة are feminine, and فكرة is one of them. Since adjectives must agree with the nouns they describe, ثانية is also feminine.

Compare:

• مشروع ثانٍ = a second project
• فكرة ثانية = a second idea

The noun changes gender, so the adjective changes too.

It can mean either, depending on context.

Literally, ثانية means second. But in many contexts, it can also be understood as another or a different.

So فكرة ثانية can mean:

• a second idea
• another idea

If you want to make another especially clear, Arabic often uses:

• فكرة أخرى = another idea

So:

• عندي فكرة ثانية = I have a second / another idea
• عندي فكرة أخرى = I have another idea

Because the phrase is indefinite.

Here, the meaning is an idea or another idea, not the idea.

In Arabic:

• no الـ usually means indefinite
• الـ makes a word definite

So:

• فكرة ثانية = a second / another idea
• الفكرة الثانية = the second idea

Notice that the adjective follows the noun in definiteness too:

• indefinite noun → indefinite adjective
• definite noun → definite adjective

It means about this project.

• عن = about / concerning
• هذا = this
• المشروع = the project

So:

• عن هذا المشروع = about this project

This phrase tells you what the idea is about.

A learner should notice that Arabic often uses prepositions a little differently from English, so it is good to memorize عن with the meaning about / concerning in contexts like this.

Because المشروع is a masculine noun.

So Arabic uses the masculine singular demonstrative:

• هذا المشروع = this project

If the noun were feminine, you would use:

• هذه الفكرة = this idea

So the choice depends on the gender of the noun:

• masculine → هذا
• feminine → هذه

In Modern Standard Arabic, the normal order is:

• demonstrative + noun

So:

• هذا المشروع = this project
• هذه الفكرة = this idea

That is the standard MSA pattern learners should use.

A careful fully vocalized form is:

عِندي فِكْرَةٌ ثانِيَةٌ عَنْ هٰذَا المَشْروعِ

A natural pause-form pronunciation is roughly:

ʿindī fikra thāniya ʿan hādhā al-mashrūʿ

A couple of helpful notes:

• عندي = ʿindī
• فكرة ends in ة, which is usually pronounced like -a in pause: fikra
• ثانية is pronounced thāniya
• هذا is hādhā
• المشروع is al-mashrūʿ

In normal spoken reading, many final case endings are not strongly pronounced unless someone is reading very carefully.

Yes, absolutely.

That version is also correct and very natural in Modern Standard Arabic:

• لديّ فكرة أخرى عن هذا المشروع

Compared with your original sentence:

• عندي فكرة ثانية عن هذا المشروع

the differences are mainly these:

• لديّ is often felt to be a bit more formal or written
• أخرى more clearly means another
• ثانية can mean second or another, depending on context

So both are good, but the version with أخرى may be clearer if you specifically mean another idea, not necessarily the second idea.