Short Answer
The language used on a generation ship would change very little over the course of the journey despite its 1,000 year duration.
Long Answer
Contrary to widespread conventional wisdom, language drift is very modest over long periods of time in the absence of specific environmental or language contact experiences the cause a language to change over time.
One of the best historical comparisons is the Icelandic language. It is almost unique in experiencing very little language contact or mass migration for a very long period of time. (In contrast, for example, Old English arose from the Anglo-Saxon invasion of a linguistically Celtic and Latin speaking island, and Middle English arose as the main modification in the wake of England's conquest by a French speaking Normal elite and England was extremely involved in contacting other civilizations globally as part of its diplomatic, trade and cultural interactions with Europe and as a result of its global empire.)
Iceland was an uninhabited island (hence not subject to any substrate influences unlike many other colonists) settled in the historic era by colonists who spoke a single shared language fairly close to "Old Norse" (the proto-language of the Germanic languages), who had a total population similar to that of a generation ship.
Somewhat more subtly, Iceland's colonists were looking for economic opportunities, rather than constituting a cultural minority fleeing persecution in their homeland the way, for example, that the Puritan colonists of New England were, so they were not motivated at the outset to intentionally set out to put linguistic and cultural distance between themselves and the speakers of their original mother tongue the way that early Americans deliberately attempted to distinguish themselves linguistically from England to assert a distinct cultural identity. The Iceland situation of a group of people not committed ideologically to putting cultural distance between themselves and their ancestor population would likely also be true of the people on a generation ship.
The descendants of the original colonists of Iceland remained in constant communication with each other ever after, just as generation ship residents would, although this did break up from a single community into about a dozen or so chiefdoms that were substantially autonomous from each other and had distinct identities although they did retain a very thin form of island-wide weak government. So, tribal divisions, per se, do not distinguish a generation ship from Iceland.
And, Iceland received almost no significant waves of migration prior to the recent past, so it has never had a large community of outside language learners to influence its linguistic development (just as would be the case in a generation ship).
While Iceland wasn't as isolated from other people as generation ship would be, for a period of time roughly equal to the length of a generation ship journey prior to the introduction of telecommunications, Iceland was very isolated from the rest of the world, and the vast majority of what little contact it did have with the rest of the world from visiting ships was with people who spoke the closest linguistic relative of their language in the world. So, it did not have any languages from which to borrow words, and it was not subject to any areal effects from neighboring languages in a significant way (apart from Danish rule whose effect is noted below).
Icelandic is by far the most linguistically static of the Germanic languages, with the only significant change being a change in pronunciation that arose during a several century period of Danish rule, a country that speaks a closely related by different language derived from Old Norse. A generation ship would not experience this outside influence and would not need to develop a new written language as Iceland did.
As Wikipedia explains (at the link above):
The oldest preserved texts in Icelandic were written around 1100 AD.
Much of the texts are based on poetry and laws traditionally preserved
orally. The most famous of the texts, which were written in Iceland
from the 12th century onward, are the Icelandic Sagas. They comprise
the historical works and the eddaic poems.
The language of the sagas is Old Icelandic, a western dialect of Old
Norse. The Dano-Norwegian, then later Danish rule of Iceland from 1536
to 1918 had little effect on the evolution of Icelandic (in contrary
to the Norwegian language), which remained in daily use among the
general population. Though more archaic than the other living Germanic
languages, Icelandic changed markedly in pronunciation from the 12th
to the 16th century, especially in vowels (in particular, á, æ, au,
and y/ý).
The modern Icelandic alphabet has developed from a standard
established in the 19th century, primarily by the Danish linguist
Rasmus Rask. It is based strongly on an orthography laid out in the
early 12th century by a mysterious document referred to as The First
Grammatical Treatise by an anonymous author, who has later been
referred to as the First Grammarian. The later Rasmus Rask standard
was a re-creation of the old treatise, with some changes to fit
concurrent Germanic conventions, such as the exclusive use of k rather
than c. Various archaic features, as the letter ð, had not been used
much in later centuries. Rask's standard constituted a major change in
practice. Later 20th-century changes include the use of é instead of
je and the removal of z from the Icelandic alphabet in 1973.
Apart from the addition of new vocabulary, written Icelandic has not
changed substantially since the 11th century, when the first texts
were written on vellum. Modern speakers can understand the
original sagas and Eddas which were written about eight hundred years
ago. The sagas are usually read with updated modern spelling and
footnotes but otherwise intact (as with modern English readers of
Shakespeare). With some effort, many Icelanders can also understand
the original manuscripts.
Also, usually new vocabulary is developed in response to new topics of conversation. Since the environment would change less and there would be fewer new things on a generation ship than in Iceland over the same time period (in part, due to a generation ship's lack of trade with the outside world that Iceland had) one would expect there to be less innovation in vocabulary on a generation ship than there was in Iceland. Still, as noted by @Hyperneutrino one might expect words that residents of a generation ship have no reason to use to be lost early on from popular speech (although presumably written sources would still preserve the old words for those so inclined, as a generation ship's residents would not be illiterate and written languages stabilize spelling and prevent old words from dying completely), and there would be a few new words developed over time.
Another clue with which I have first hand experience, which corroborates the example of Iceland, is the development of motherland languages in immigrant communities. For example, the dialect of Korean spoken in Korean immigrant communities in the U.S. that formed shortly after the Korean War is much more similar to the dialect spoken at the time of migration than the dialect of Korean currently spoken in South Korea, notwithstanding the fact that there have been many visits to South Korea and there has been access to modern Korean television and music in that time period in the U.S. Small, relatively isolated communities are simply much less linguistically innovative than large communities that are strongly connected to neighboring communities that speak different languages.
In summary then, it is likely that the language of a generation ship would be extremely static over a period of approximately 1,000 years, because generally speaking it would be in conditions very similar to those of Iceland which historically has had an extremely static language over a similar time period under similar conditions. And, several of the factors that are know to have led to the modest changes between Old Icelandic and Modern Iceland over that time period would be absent on a generation ship.