Why did Samara contact Aiden?

TL;DR:

It's likely from his father, although this explanation comes from the Japanese movies (not the original novels) and isn't elaborated on in the American films.

Long Answer

It's not really explained in the movie, but there is some information given in the original Japanese materials:

Original novel:

The Ring evolved from Ring, a 1991 Japanese horror movie. In that, the equivalent of Samara is Sadako Yamamura, with Aiden's equivalent being a girl called Yoko Asakawa, the one year old daughter of Shizuka and Kazuyuki Asakawa. Kazuyuki is the protagonist, along with his friend, Ryuji Takayama.

In these stories, Samara is shown to have a number of powers, including ESP, telekinesis, healing abilities and the power to the make the video tape. However, despite having some awareness of paranormal activities, none of the other characters have any obvious psychic or supernatural powers.

So, in summary:

   Shizuka - - - - - - - - - - - Kazuyuki                     Ryuji
                    ||
                    ||
                   Yoko  

-- = married to.   
|| = child of.

Original films/series:

The Ring was then made into a few films, including Ringu, a 1998 Japanese horror film. In this film, the characters are the same as the book, but with some important subtle differences. Ryuji Takayama is still the friend of the protagonist, but importantly, he's also the ex-husband of the protagonist's wiife, Shizuka. He has some near-psychic powers which are increase significantly in Ring: Final Chapter, an original TV series based on the movie from 1999. In that series, he's effectively a guru of the supernatural. This is used in to effectively cut corners in the story and allow the Ryuji to discover things that took much longer to unearth in the books.

Now, as already mentioned, in this film he is the ex-husband of Shizuka. This is important. The character of Yoko, the daughter in the books, is replaced with that of Yoichi Asakawa, a seven year old boy. He is not the son of Kazuyuki like in the books, but rather the son of Ryuji. He has clearly inherited some psychic abilities from him, including ESP and clairvoyancy.

So, in summary:

   Ryuji / / / / / / / / / Shizuka - - - - - - - Kazuyuki          
                 ||
                 ||
               Yoichi

// = divorced from
-- = married to.   
|| = child of.

American films:

In The Ring, the 2002 American remake of Ringu, the character of Yoichi Asakawa has become Aiden (from your question). It appears the American filmmakers ported over some of the psychic abilities of Yoichi to Aiden and this is the best explanation I can find as to why he has a connection to Samara (or at least, why he's partly psychic).

Now, this is by no means a satisfying solution. EntertainmentScene360 put it well, when they commented:

The Ring takes the character of Yoichi Asakawa from Ringu, the reporter's quiet young son who doesn't have an awful lot of lines in the movie into Aiden Keller who is the same boy, however, this version of the character appears to have the psychic ability of Ryuji from the original and shares a sort of link with Samara, the American's little girl version of the vengeful spirit.

In other words, whilst Yoichi has some psychic powers, this connection with Samara appears to be something totally new.

Conclusion:

There is another thing I didn't mention. Sadako Yamamura, the original antagonist, was a young woman (at least 19 years old). The American version features Samara, an eight year old girl.

I think this change was made as it has always been unsettling to see children in horror films, portraying unusual and bizarre characters (think The Shining, Children of the Corn, Omen, The Exorcist, The Sixth Sense etc). I would then suggest that Aiden's new connection with Samara is done to increase this feeling and have two unsettling, supernatural characters.

I'm aware I haven't really explained why Samara contacted Aiden (which I just don't have an explanation for), but hopefully it gives some understanding as to where his psychic abilities could have originated from.