This is a second/supplementary answer, bringing newly-published data relevant to the question
"Why are scientists saying that the Omicron COVID-19 variant is a reason to get a booster?" and in particular, why "the proposed solution is extra doses of the same old vaccines" (when initial courses of those original vaccines have shown clear signs of relative ineffectiveness against this new variant).
My earlier answer (Dec 11) tried to describe the part of the normal immune system that enables a booster vaccine dose to generate a diversified immune response, through the intervention of the random-mutation part of the system.
While this describes an in-principle mechanism for booster effectiveness against Omicron in the current scenario, descriptions of mechanism inevitably leave aside an important point : 'how effective' can such booster vaccine doses be?
To answer that, as the old saying goes, "the proof of the pudding is in the eating". But with the Omicron variant so new, the main scientific results until the last few days had to be based on laboratory-bench testing of blood-serum/antibody samples.
Naturally of great interest is actual clinical data, and within the last week, the original laboratory data have been supplemented by publication of a (preprint) report of a UK-based case-control study surveying actual patients.
Again, since the Omicron variant is still so new, and the Omicron-caused diseases still so recent and perhaps in early stages, the new study is "unable to determine protection against severe forms of disease, due to the small number of Omicron cases so far, and the natural lag between infection and more severe outcomes". Nevertheless, the new results do show that the indications from the laboratory antibody studies are being confirmed in the early clinical experiences of covid-19 due to Omicron, i.e. the results are showing that booster dosing with the Pfizer-Biontech vaccine gives a marked improvement in protection against the Omicron-caused disease seen so far compared with the effects of either of the previous 2-dose vaccine programs alone.
The report is still too recent to have been refereed, but can be read at "Effectiveness of COVID-19 vaccines against the Omicron (B.1.1.529) variant of concern". The authors come from hospitals and research institutions in London, Oxford and Cambridge plus the national health-security agency. The data analysis was enabled by the UK National Immunisation Management System with details of covid vaccinees.
The authors' introduction cites previous "preliminary neutralisation data" (that is, laboratory studies of antibody inactivation of virus) against the Omicron variant.
"The South African and German studies, as well as unpublished data
from the UK, indicate a 20- to 40-fold reduction in neutralising
activity in sera from 2-dose [Pfizer/Biontech] vaccine recipients
compared to early pandemic viruses, and at least a 10-fold reduction
compared to the Delta variant." Thus the "effectiveness [of available
vaccines] against symptomatic disease with the Omicron variant is
significantly lower than with the Delta variant."
Among those who had received 2 doses of [Pfizer/Biontech], vaccine
effectiveness [against Omicron] was 88.0% ... 2-9 weeks after dose 2,
dropping to 48.5% ... at 10-14 weeks post dose 2 and dropping further
to between 34 and 37% from 15 weeks post dose 2.
Vaccine effectiveness increased to 75.5% ... after the booster among
those who had received [Pfizer/Biontech] as the primary course."
"Among those who had received 2 doses of [Oxford/AstraZeneca], there
was no protective effect of vaccination against symptomatic disease
with Omicron from 15 weeks after the second dose.
Among those who received [Oxford/AstraZeneca] as the primary course,
from 2 weeks after a [Pfizer/Biontech] booster dose, vaccine
effectiveness [against Omicron] increased to 71.4% ... .
"All of the Omicron estimates are subject to significant uncertainty with wide confidence intervals", because of the low number of patients with identified Omicron-caused disease (just under 600 patients between 27 November when Omicron patients began to be identifiable and 9 December when the data were obtained, although Delta patients and test-negative controls in the data came from a larger time-interval and numbered over 56,000 and over 130,000 respectively.
Thus the new study concludes that
"a booster dose of [Pfizer/Biontech] following either
[Oxford/AstraZeneca] or [Pfizer/Biontech] as a primary course" brings
"moderate to high vaccine effectiveness against mild infection [due to
Omicron] of 70-75% ... in the early period after [the booster dose]."
By contrast, for patients without booster doses,
"Our findings indicate that 2 doses of vaccination with
[Pfizer/Biontech,] or [Oxford/AstraZeneca] are insufficient to give
adequate levels of protection against infection and mild disease with
the Omicron variant, although we cannot comment on protection against
"Booster doses of [Pfizer/Biontech] provide a significant increase in
protection against mild disease and are likely to offer even greater
levels of protection against severe disease. As such our findings
support maximising coverage with third doses of vaccine in highly
vaccinated populations such as the UK. Further follow-up will be
needed to assess the duration of protection of booster vaccination."
So the boosted patients in the study experienced very much better protection against Omicron than patients who had only received two doses, whether of Oxford/AstraZeneca or Pfizer/Biontech.
In the US, in reaction to recent results, "Dr. Anthony S. Fauci said Wednesday [Dec 15] ... that “at this point, there is no need for a very specific booster” designed especially to fight Omicron." New York Times.