It's an age-old question, and is one that is usually answered a lot less well than it should be. As is usually the case for questions like this, the truth is much more complicated than a simple yes or no answer, and I hope to shed a little bit of light on the issue below.
In my view, the best way to think about money is to consider it a tool, and like all tools, it can be put to good use, wasted, or used counter-productively (or even for malevolence). The value of a tool depends on who wields it and how they wield it, and so does the level of happiness deriving therefrom.
The most valuable thing money can do for you on a personal level, other than meeting your basic survival needs (a roof over your head; providing access to healthy food, etc) and liberating you from worrying about your continuing ability to meet those needs, is to provide you with freedom and options about how you spend you time. Time is the most valuable resource there is in life, and although money can't buy more of it in a literal sense, it can buy more quality time (by substituting low-quality time - namely time committed to undertaking activities you would rather not do - for value-add time - time spent doing the things you enjoy and that can make you happy).
If someone's true passion, for instance, is a creative but under-remunerated pursuit - perhaps building a YouTube channel (or writing a blog for that matter) - money can provide that individual with the flexibility to commit their energies to that task, unburdened with the concern about how to make next month's rent. That individual will be happier pursuing that task full time than trying to squeeze it into evenings and weekends with left-over energy, while spending the bulk of their productive time working an unfulfilling job to pay the bills. Money used in this manner is an extremely liberating thing, and it will indeed help make a person much happier.
However, if one is able to find a way to pursue their passion while making a decent enough living as part of the process (though being far from rich), then additional money is unlikely to make much difference to the happiness of the individual. They are already living in the way they would like to live, and their consciousness is already occupied in a meaningful way (being happy requires being in a state of flow, which requires that consciousness be occupied by meaningful work/activities; when a psychic vacuum emerges, that is when corrosively negative emotions have room to creep in). The only value money would have for this individual would be to guard against a future change in external circumstances that would not allow the prior happy state of affairs to be perpetuated.
Where money will not make someone happy - or even make them unhappier - is if they attempt to use it to compensate for deeply-rooted psychological demons or other character flaws, or use it to pursue an unhealthy degree of pleasure/hedonism. For instance, someone with financial resources who feels inadequate and insecure, and therefore feels the need to seek other people's approval to an unhealthy degree, might seek to do so through conspicuous consumption. That is likely to lead to nothing more than further frustration and bitterness, as other people are likely to simply resent or distrust the person (people don't like other people's success flashed in their faces, because all it does is make them feel less successful themselves), and the lack of ability to win people's approval this way could well result in the said wealthy-but-unhappy person's feelings of inadequacy becoming even more entrenched. And the reassuring belief that 'if only I had more money, I would be happier' would not be available to this person, which could result in a greater feeling of hopelessness and despair.
Money has the capacity to amplify. It can amplify the good and virtuous in people, and it can also amplify the bad and malevolent. If somebody is psychologically healthy, money is likely to do nothing but add value to their life. But if someone is psychologically unstable, it is likely to do little more than hasten their spiral into the abyss. Rich drug addicts have the power to kill themselves faster than poor ones, as they can afford to indulge their vice to a much greater extent than the fiscally constrained can. Rich people can also afford to be obnoxious or arrogant as they do not need to defer to others (e.g. a boss) to secure their financial wellbeing. However, this sort of attitude can lead to them alienating themselves from people, and meaningful relationships with others are an important part of happiness. Someone who is an unpleasant person but who is rich, is likely to become even more unpleasant and distasteful to others, and become unhappier as a result.
Being rich also has the ability to provide people with a degree of freedom in their life that some may not be able to cope with. Human beings need structure in their lives, as it is a way of imposing 'order on chaos', and keeping the psychological turmoil that can result from a sufficient lack of order at bay (read/listen to the brilliant psychologist Jordan Peterson for more on this). Working 9 to 5, Monday to Friday, provides structure and a daily routine to many people's lives, as well as a source of meaning. They may fantasise about being so rich they wouldn't need to work for a living, but thrown into that position, many would struggle to find a new structure for their day, and could wind up miserable.
Studies have shown that people are actually less happy, on average, when they are on vacation/travelling than when engaged in their daily work ritual. The lack of order/structure is why. The myth that being rich enough to lie on a beach in a tropical destination every day is the very definition of eternal human bliss is just that - a myth. People would find themselves miserable doing that much more rapidly than they would believe.
The fabulous movie The Shawshank Redemption discussed how life-long inmates can become 'institutionalised'. When released back into the free world, many could not cope with the lack of structure. There was too much chaos and not enough order. There are shades of this for the working classes. Many would struggle without the order provided by a structured work day. This is particularly the case because our whole society and social structures (outside of school/university) are based around work. Being outside of this system could feel isolating/disorientating to some people.
Wealthy people have this freedom, and - should they choose - the availability of vast amounts of unstructured time, and if they do not find a way to use it intelligently, it could actually conceivably make them less happy. However, it is not the money itself that would be making these folk less happy - it is the fact that the power and choices the tool is providing are not being put to good use.
Another way money can make people unhappy is the risk of its subsequent loss. Cher once argued that it was better to have loved and lost than to have never loved at all. I won't offer an opinion on that, but I will say I would disagree if the analogy were to be extended to money. I think it is better to have not been rich at all than to have been rich and then to have become poor again. Having money therefore also has the capacity to cause unhappiness due to its perpetual risk of loss. This is why you sometimes see formerly-rich 50yo bankers throwing themselves out of windows or in front of trains during financial crises. They have committed their entire lives to accumulating money, and the prospect of losing it all and having to start over again in their 50s is too unbearable to face (particularly because their entire identity and sense of self-worth is tied to their balance sheets).
Conflations cause misunderstandings
Part of the misunderstanding about the question of whether money makes you happy stems from the conflation of 'money' with 'conspicuous consumption' and/or 'a high salary job'. The ability to indulge in the consumption of lavish goods and other luxuries is only one of the things access to financial resources provides, and it is the most superficial and unfulfilling. The reason is simple: there is a fundamental difference between pleasure and happiness. Money spent on lavish consumption increases one's ability to procure pleasure, but everyone knows that the pursuit of pleasure does not lead to sustained happiness, and it is no different for the wealthy.
However, money can do so much more than enable obscene over-consumption. Beyond the personal self-actualisation benefits it can provide by buying you the time and freedom to pursue your passions (already discussed), it can also give you the power to support charitable causes you believe in. My charitable budget is not that large, but I use what I do have to support YouTube creators and advocates for causes that I believe in (usually via Patreon). That brings me happiness, and I hope to be able to do much more in this regard in future years, if my financial resources permit.
Money also gives you the power to mitigate suffering and misfortune in the world when you encounter it - either amongst your family or friends, or in the wider world. Many people like to pray for the suffering. I view that as a means of making yourself feel like you're doing something to help, when in actual fact you are doing nothing at all. Money gives you the actual capacity to help, and to help meaningfully, those you care about or those in need, and this can also lead to happiness.
The other major conflation is confusing 'money' with 'a high salary job'. The two are absolutely not the same. High-paying jobs often come attached with long hours and a lot of stress. You get paid more partly because you give up more - in terms of time, energy, and health. What is given up in the pursuit of a high salary - sleep; sufficient exercise; social time with friends and family; and sufficient time and energy to pursue one's passions - are all sacrifices that can be reasonably expected to corrode one's happiness. Furthermore, if the additional funds earned are wasted on keeping-up-with-the-Jones' conspicuous consumption, little will be gained, rendering this choice of lifestyle highly likely to lead to unhappiness (unless you intrinsically love your work, and/or are doing it merely for a period of time to save enough money as fast as possible to finance you in to your next venture).
This is particularly the case for high-income folk that get themselves into a lot of debt (e.g. large mortgages on expensive houses), because worse than this lifestyle alone is the sense of entrapment that comes from feeling like there is no choice but to continue living that unrewarding life. After adapting to a high-income, high-consumption, high-debt lifestyle, they feel that there is no choice but to perpetuate it.
This absolutely can and does make a lot of people unhappy, and I suspect this is what a lot of people mean when they say 'money doesn't make you happy'. However, it is not money that is making these folk unhappy, but rather the excessive pursuit of money (or more specifically, conspicuous consumption) to the detriment of more important values in life - health; relationships; and meaningful work/hobbies/interests.
Ultimately, it is a question of character and individual choices. Money is merely a tool, which if used wisely, can be a lush oasis that is capable of facilitating one's pursuit of happiness, but if used or chased in an ill-consider manner, can turn into a toxic mirage.