Q:

The best choice for Person table primary key

What is your choice for primary key in tables that represent a person (like Client, User, Customer, Employee etc.)? My first choice would be an social security number (SSN). However, using SSN has been discouraged because of privacy concerns and different regulations. SSN can change during person lifetime, so that is another reason against it.
I guess that one of the functions of well chosen natural primary key is to avoid duplication. I do not want a person to be registered twice in the database. Some surrogate or generated primary key does not help in avoiding duplicate entries. What is the best way to approach this?
What is the best way to guarantee uniqueness in your application for person entity and can this be handled on database level with primary key or uniqueness constraint?

A:

I don't know which Database engine you are using, but (at least with MySQL -- see 7.4.1. Make Your Data as Small as Possible), using an integer, the shortest possible, is generally considered best for performances and memory requirements.
I would use an integer, auto_increment, for that primary key.
The idea being :

If the PK is short, it helps identifying each row (it's faster and easier to compare two integers than two long strings)
If a column used in foreign keys is short, it'll require less memory for foreign keys, as the value of that column is likely to be stored in several places.

And, then, set a UNIQUE index on an other column -- the one that determines unicity -- if that's possible and/or necessary.

Edit: Here are a couple of other questions/answers that might interest you :

What’s the best practice for Primary Keys in tables?
How do you like your primary keys?
Should I have a dedicated primary key field?
Use item specific prefixes and autonumber for primary keys?

A:

As mentioned above, use an auto-increment as your primary key. But I don't believe this is your real question.
Your real question is how to avoid duplicate entries. In theory, there is no way - 2 people could be born on the same day, with the same name, and live in the same household, and not have a social insurance number available for one or the other. (One might be a foreigner visiting the country).
However, the combination of full name, birthdate, address, and telephone number is usually sufficient to avoid duplication. Note that addresses may be entered differently, people may have multiple phone numbers, and people may choose to omit their middle name or use an initial. It depends on how important it is to avoid duplicate entries, and how large is your userbase (and thus the likelihood of a collision). 
Of course, if you can get the SSN/SIN then use that to determine uniqueness.

A:

What attributes are available to you? Which ones does your application care about ?  For example no two people can be born at exactly the same second at exactly the same place, but you probably don't have access to that data at that level of accuracy!  So you need to decide, from the attributes you intend on modeling, which ones are sufficient to provide an acceptable level of data integrity.  Whatever you choose, you're right in focusing on the data integrity aspects (preventing insertion of multiple rows for the same person) of your selection.  
For Joins/Foreign Keys in other tables, it is best to use a surrogate key.
I've grown to consider the use of the word Primary Key as a misnomer, or at best, confusing.  Any key, whether you flag it as Primary Key, Alternate Key, Unique Key, or Unique Index, is still a Key, and requires that every row in the table contain unique values for the attributes in the key. In that sense, all keys are equivilent.  What matters more (Most), is whether they are natural keys (dependant on meaningful real- domain model data attributes), or surrogates (Independendant of real data attributes)  
Secondly, what also matters is what you use the key for..  Surrogate keys are narrow and simple and never change (No reason to - they don't mean anything) So they are a better choice for joins or for foreign Keys in other dependant tables.  
But to ensure data integrity, and prevent insertion of multiple rows for the same domain entity, they are totally useless...  For that you need some kind of Natural Key, chosen from the data you have available, and which your application is modeling for some purpose.
The key does not have to be 100% immutable.  If (as an example), you use Name and Phone Number and Birthdate, for example, even if a person changes their name, or their phone number, you can simply change the value in the table.  As long as no other row already has the new values in their key attributes, you are fine.  
Even if the key you select only works in 99.9% of the cases, (say you are unlucky enough to run into two people with the same name and phone number and were coincidentally born the same day), well, at least 99.9% of your data will be guaranteed to be accurate and consistent - and you can for example, just add time to their birthdate to make them unique, or add some other attribute to the key to distinquish them.  As long as you don't have to update data values in Foreign Keys throughout your database because of the change, (since you are not using this key as a FK elsewhere) you are not facing any significant issue.

