Seeking guidance for a sentence that we are going to use in creating a facebook page

These three are very near to each other but the second one 'Daily property updates' is the correct one and is conveying the meaning effectively. It is grammatically correct also. You can use it for your page.
If by chance this page name is not available and already under use by someone else and you do not like the repetition as repetition may reduce the traffic to a particular page, then in that case you can try the other catchy names like 'Property updates 24/7' and many similar combinations.

In the three combinations you have given, 'Daily Property Updates' is correct. There are no grammar mistakes and also all the spellings are correct. You can think of using this.
If you use it it gives a feeling that you will update the information once a day. Instead of that if you say " Latest Property Updates", one will get a feeling that it is the latest. So I feel that will be more appealing to the users. If I were you I will go with my suggested name. You can decide as per your choice.

While going through your query regarding the three combinations as stated by you, I find first two combinations are devoid of grammatical errors and can be safely be used.
The first combination signifies that your cousin is engaged with the dealings of different properties considering their different areas and their existence in the different localities. In that way, your cousin is giving a detailed message of the properties of different combinations to attract the interested customers. In that situation the first combination fits most.
Daily Property Updates signifies that your cousin would be more interested to add the latest feature for which there is the latest updates of the parameters of the same property in terms of its area and its cost angle.
I personally feel both the two combinations as suggested from my end are correct.