The Hobbits of the Shire and from Bree adopted the common tongue one thousand years before the events of the war of the ring. But they used this tongue in their own manner and with carelessness, although those more instructed were able to speak more formally when the occasion required it.
There is no record about a specific tongue for Hobbits and it seems that since early days they used the tongues of nearby men tribes so, it was a matter of time for them to adopt the common tongue and by the time they settled in Bree they had almost forgotten their previous tongue.
Despite this, in the times of Frodo and the ring, the hobbits still had some words from their previous adopted tongues (most likely a Rohirrim-ish tongue) and the likes of Rohan. The most notorious examples for this are the names they used for days, months and seasons, but also the names they used, which resembled more to the ancient tongues used in Rohan rather than to the common tongue of the middle earth during those times.
Men used to call them halflings and Elves called them Perianath, but the name Hobbit, the official in their own accord, had lost its meaning centuries before and no one knows about its origins. There are speculations about the name though, and it seems that between Hobbit races, it was a name used by The Stoors and the Fallohides to name the Harfoots with a combination of the Rohan word for hole-diggers, holbytla.
