[Lalwendë]

Here's what I found many moons ago in Appendix F:

Of these things in the time of Frodo there were still some traces left in local words and names, many of which closely resembled those found in Dale or in Rohan. Most notable were the names of days, months, and seasons; several other words of the same sort (such as mathom and smial) were also still in common use, while more were preserved in the place-names of Bree and the Shire. The personal names of the Hobbits were also peculiar and many had come down from ancient days.

There is no record of any language peculiar to Hobbits. In ancient days they seem always to have used the languages of Men near whom, or among whom, they lived. Thus they quickly adopted the Common Speech after they entered Eriador, and by the time of their settlement at Bree some had already begun to forget their former tongue. This was evidently a Mannish language of the upper Anduin, akin to that of the Rohirrim; though the southern Stoors appear to have adopted a language related to Dunlendish before they came north to the Shire.

Hobbit was the name usually applied by the Shire-folk to all their kind. Men called them Halflings and the Elves Periannath. The origin of the word hobbit was by most forgotten. It seems, however, to have been at first a name given to the Harfoots by the Fallohides and Stoors, and to be a worn-down form of a word preserved more fully in Rohan: holbytla 'hole-builder'.

We also have the similarities in word roots such as Hama and Hamfast. The store of old Rohirric legends about the Hobbits. And the similarities in funerary practices found on the Barrow-Downs and in contemporary Rohan.

Note that Tolkien said there was no record of any specific Hobbitish language. This will likely have been a non-literary culture so no written records remain - like Celtic languages such as the Cymric spoken in Rheged, traces remain in familiar, everyday words such as those used to name rivers and villages - and the Hobbits brought this language with them when they travelled West.

The origins given for the word Hobbit are also revealing as they do not come from anything relating to their size, they come from where they choose to live. At some point, this branch of Men began living in holes - did this lead to their growing smaller over time? Note that it was the hole-building which was distinctive, and not their height. Couple this with the Elves not identifying them as somehow a distinctive people until after they begin to take on a particular appearance, it all points to a common ancestry somewhere along the line.