Huh? None of the Hobbits really had rank.  Even if we take Frodo as Esquire and assume his cousins are of equal rank (if esquire even probably means diddly to Hobbits) Sam's still just a Gardner. Gimli while not heir apparent still has a good solid claim to the throne. 

So Dain II Ironfoot is king.

Heir apparent is his son Thorin III Stonehelm 

Since I couldn't find a DOB for Durin VII (Thorin III Stonehelm's son)...I'm going to assume he wasn't born yet

Next would be Balin but he's dead....granted they don't know it at the time...

So Dwalin is 2nd in line for the throne

After Dwalin would be Oin...but Oin is also dead so...

Gloin is 3rd in line and Gimli is his heir

4th isn't an insignificant spot...especially in times of war (something about rulers personally leading the troops does that).  It's certainly higher than Boromir being son of Denethor as Stewards are ultimately just high ranking servants.

That said the Fellowship was a group of **VOLUNTEERS** and weren't chosen by any save themselves.