You ask a good question and I'm sure you're not the only one in a situation like this. Unfortunately, I don't think that anyone here could provide an exact answer for you, or anyone in a similar situation. The best way to figure it out would be to ask your mother/relative/friend to have an open communication about vaccines. Ask where they got their information, but don't rush to judgment. People's brains can be tricky and we're not always logical, so give it some time and try to establish trust. Not only could trust and communication with you allow you to figure out why mom's afraid, but trust and communication with a doctor might help alleviate the fear too.
When mothers are thinking about vaccinating (or not vaccinating) their children, trust in a doctor is important (1). Agreeing to vaccinate oneself is obviously not the same thing, and there are plenty of possible complications these days. But especially considering how important sources of information are, getting information from a trusted doctor could be one way to alleviate the fear. I'll link to some layers here that might help start a conversation about vaccination with someone. This is not comprehensive in scope or depth.
At a super basic level, some people have a fear of needles (2). This might be related to a tendency to protect the body, or a fear about introducing a foreign presence into the body. You can think about the body as a barrier that you generally don't want to have penetrated. Fears related to blood-injection-injury have been linked to aversion (that is, disgust) toward body envelope violations (3). [Note. I have not found empirical data supporting or refuting a direct link between fear of needles and vaccine hesitancy.]
On top of the complex psychology around an injection itself, vaccines commonly inject virus to prevent future virus, which is not the most intuitive idea (wikis: inject cowpox to prevent smallpox disease, inject inactivated flu to prevent flu illness). These emotional/cognitive components could contribute to fear or hesitancy of vaccination, and could be amplified by social factors.
Anti-vaccination attitudes (which could be related to fear of a vaccine) are more complicated because they also involve worldviews, such as conspiratorial thinking (4) and social norms (5). One possible (albeit nonspecific) way to think about this is through the idea of "groupthink", a phenomenon in which people are able to come to consensus under certain conditions, including limited search and appraisal of information, insulation from experts, and directed leadership (6). There was one RETRACTED paper which reported that vaccines might be related to autism in the late 1990s and while the evidence did not support the idea, it was/is a topic in the anti-vaccine movement (retraction, rebuttal (7), more commentary from that time (8)). Here's (9) a point of view which talks about some influential figures in the anti-vaccine movement before 2017.
Just to be explicit, this answer is not specific to the COVID-19 vaccine. In fact, I only referenced material prior to the beginning of the pandemic. There are probably many reports of COVID-19-specific vaccine hesitancy; I would be surprised if they came to substantively different conclusions than the past work (in that getting information from a trusted doctor is probably a big factor in alleviating fear and hesitancy of vaccine). The social/political climate could be a larger factor these days, especially as politics influences sources of information and trust. Social media could also be important (again, especially as a source of trusted information for many people).
If you're looking at the cited material, I might suggest starting with references one (Benin et al., 2006), five (Kestenbaum & Feemster, 2015), and four (Hornsey et al., 2018).
Benin, A. L., Wisler-Scher, D. J., Colson, E., Shapiro, E. D., & Holmboe, E. S. (2006). Qualitative analysis of mothers' decision-making about vaccines for infants: the importance of trust. Pediatrics, 117(5), 1532-1541.
McLenon, J., & Rogers, M. A. (2019). The fear of needles: A systematic review and meta‐analysis. Journal of advanced nursing, 75(1), 30-42.
de Jong, P. J., & Merckelbach, H. (1998). Blood-injection-injury phobia and fear of spiders: Domain specific individual differences in disgust sensitivity. Personality and Individual Differences, 24(2), 153-158.
Hornsey, M. J., Harris, E. A., & Fielding, K. S. (2018). The psychological roots of anti-vaccination attitudes: A 24-nation investigation. Health Psychology, 37(4), 307.
Kestenbaum, L. A., & Feemster, K. A. (2015). Identifying and addressing vaccine hesitancy. Pediatric annals, 44(4), e71-e75.
Turner, M. E., & Pratkanis, A. R. (1998). Twenty-five years of groupthink theory and research: Lessons from the evaluation of a theory. Organizational behavior and human decision processes, 73(2-3), 105-115.
Taylor, B., Miller, E., Farrington, C., Petropoulos, M. C., Favot-Mayaud, I., Li, J., & Waight, P. A. (1999). Autism and measles, mumps, and rubella vaccine: no epidemiological evidence for a causal association. The Lancet, 353(9169), 2026-2029.
DeStefano, F., & Chen, R. T. (1999). Negative association between MMR and autism. The Lancet, 353(9169), 1987-1988.
Smith, T. C. (2017, July). Vaccine rejection and hesitancy: a review and call to action. In Open forum infectious diseases (Vol. 4, No. 3). Oxford University Press.