Hah! Can't believe I forgot to necro-bump this thread once we found out about the Shinju, and where the Rinnegan originated from. Well, I think it'll become relevant again once Madara starts revealing his Rinnegan usage.
And I am not sure why so many believe the Uzumaki's were closer to the Sage. It is told perfectly to us that the Senju and Uchiha are the closest decedents. The Uzumaki's were related to the Senju but it is plainly obvious that the Senju were closer. Notice it was Hashirama's cells, not a Uzumaki's that were needed to form the Rinnegan.
Eh? It's not like Madara stole some Uzumaki cells first, failed to get Rinnegan, then stole Hashirama cells, and succeeded.
Would it also work with the cells of Hashirama's aunt, who does most of the cooking and tidying? Just forget about Senju clan, forget about Uchiha, Hashirama and Madara were just two incidental freak occurances within a shitload of people that are of Sage lineage. Could've just as easily been an Uzumaki and a Hyuuga (considering the Uchiha branched off later than the Hyuuga did).
Also, what does closer mean? Are you more closely related to your grandma than your nephew? Even if you seem to have more similar characteristics, it has nothing to do with being closer.
In a less scientific sense, Uzumaki seem to be closer based on the fact that the now supposed true successor is an Uzumaki. Hashirama is just a strong guy with fortunate dominant genes, no hint or whatever of being the Sage's divine successor. Seems to me that the universe or whatever considers Uzumaki to be the core lineage.
The Senju and Uchiha are said to be the closest descendants because Kushina told us the Uzumaki was a cousin or sister clan of the Senju, she did not mention the Uchiha. Madara wanted Hashirama's cells because it is extremely potent. Just look at how powerful the person becomes - Yamato could suppress the Kyuubi, Tobi could take damage and likely spam his Mangekyo without getting blind as well as control the Kyuubi, and Orochimaru and/or Kabuto were able to control the Edo Tensei better and revive them with more strength than they would be.
I don't think it's necessarily Hashirama's cells that's required, but his DNA is just that powerful that anyone would want it regardless. Even if you don't unlock the Rinnegan, you can get other powers.