I am more interested in documenting the responses to this rather than forming a solid argument!

Opinions are extremely fickle, despite the size of your metaphorical walls, and could be changed by another opinion or objective truth at any moment. An opinion backed up by significant evidence is most definitely more powerful than one that is not.

There is no baseline for an opinion besides something that you believe to be true at the time you express it. Undermining opinions even further, if someone doesn’t understand an opinion on the same level as the person holding it, what would make the opinion matter? It’s just hot air at that point.

On a larger level, free speech is suppressed every day. Opinions that are working towards the greater good are often pushed away swiftly. I do wish all opinions did matter – combating the bad ones seems obvious, but there is no construct or method to help.

I think all opinions do matter because each one reveals something about the person that formed it. The polarisation of opinion that can be seen today happening across the world, is thought to be partly due to aspects like algorithms in smart devices feeding people the same type of political content on a loop.

To me, opinions from people of different demographic backgrounds, all matter because they can help us understand what factors influence people to think certain ways. Only by treating all opinions as if they matter can you map out what things make people form the opinions in the first place.

Voltaire said 'I disapprove of what you say, but I will defend to the death your right to say it'. This quote encompasses one's freedom of speech. Although your argument does not suggest the banning of free speech but instead the value of all speech, I see huge connections between the two. If someone expresses their opinion, however it is not recognised or considered to matter, then expression in the first place is redundant. The expression of an opinion entails communication which is immediately shut down if no one deems it to have any value. Richard Dworkin appreciates free speech, at its most basic level, as the liberty to communicate.

John Stuart Mill recognised that freedom of speech is necessary for personal growth, the fulfilment of the progressive interests of humankind and the pursuit of truth which is valuable as a precondition for knowledge.

Some opinions may be widely held to be more valuable than others or even more 'correct' that others, however this does not mean that inferior opinions do not matter. All opinions help society and people to improve, both in intellect, morality and happiness.

All opinions matter because even if they are ill informed or show inbuilt prejudice they give an insight into the person holding them.
This allows you the choice of disagreeing or ignoring but it helps form your judgement of their stance.
Opinions may vary over time with life experience but everyones' view is valid.

TL;DR - All opinions do matter. They have to matter. Especially, since we form opinions all the time on a daily basis. Sometimes consciously, sometimes unconsciously. What’s more, an opinion is born from one's own experiences and beliefs lending proof to the fact that our perceptions greatly influence our opinions.

All opinions matter, in my opinion - at least on a certain level. Admittedly, to different people, in different situations and at different times. But they matter, as they are an integral part of our everyday language in our speaking, thinking, and in our relationships and dealings with each other.

The Merriam Webster dictionary defines the word opinion as, a view, judgment, or appraisal formed in the mind about a particular matter. Additionally, it goes on to say that an opinion can also be defined as a generally held view or a belief stronger than impression and less strong than positive knowledge. The website, www.differencebetween.com confirms Merriam Webster’s definition of what an opinion is, while also explaining to us that perception is the way in which one thinks - how one looks at and understands things. As a result, the way in which one thinks about one’s opinion is influenced by one’s perception.

From here then, it's clear that opinions are not based on facts. It is influenced and based on our own perception(s) of what we believe to be true (on not) regarding a certain topic/subject matter. So opinions are subjective beliefs. And while some people are of the opinion that an opinion is just an opinion because it’s not based on facts (objective) - hard and solid evidence, opinions are often viewed as "less." Or they are not seen as accurate or truthful.

It’s important to remember that our subjective experiences also contribute, shape and influence our environment and the world around us. This includes all types of opinions, whether they are shared, not shared, impartial, etc.

So here then, is undeniable proof that no matter what the subject (person) or subject matter is that is under discussion, a person’s opinion will always matter and always carry value. Particularly, since a person’s opinion is their unique perception of a particular subject (person) or subject matter (topic under discussion).

Sources:

1. Definition of OPINION
   https://www.merriam-webster.com/dictionary/opinion [accessed February 2, 2021].

2. Difference Between Opinion and Perception
   https://www.differencebetween.com/difference-between-opinion-and-vs-perception/ [accessed February 2, 2021].

tldr; My favorite phrase: it depends

"Not all opinions matter" could also be said "some opinions don't matter." My immediate response is "who decides" and "when or in what context?"

What do we mean when we say matter? Is it that all opinions are important? Relevant? Informed?

In the context of public discourse (social media, the news, politics, etc), my opinion (ha!) is that society collectively decides which opinions matter though a constant process of approval/disapproval, promotion/demotion. But this is still a vague response to a vague claim.

To actually agree/disagree with this claim, I would need a bit more clarification, otherwise I would say some opinions matter more in some debates while marginalized opinions would matter more in others.

A knowingly uninformed opinion is an insult to everyone who hears it.

TL;DR -- There are plenty of situations in which a particular person's opinion just doesn't matter.

A pedophile's opinion on child rearing does not matter to me.

A rapist's opinion on rape victims' rights does not matter to me.

A murderer's opinion on the death penalty does not matter to me.

Idk, maybe I'm in the minority here. Most opinions matter, but some people lose their right to have opinions on certain things when they commit unspeakable acts of cruelty against others.

There are also other, less extreme instances where a person's opinion might not matter.

I don't want a politician's opinion on what works in the classroom. They have no experience in children's education, and have a severe disconnect from the diverse cultures, backgrounds, and struggles of children in today's classrooms.

I don't want a plumber's opinion on my cardiovascular health. Maybe there's a plumber or two out there who went to medical school, but I'm going to posit an educated guess that 90% of them aren't qualified to tell me what makes my ticker tick.

I don't want a doctor's opinion on what that funny noise my car is making might be, nor how they would fix it. Chances are their best guess would result in my engine blowing up.

I'm not saying that people can't have multiple talents / areas of expertise. But as a general rule, we seek out the opinions of specialized experts because they are experts in their fields. We don't go to an expert of one thing and expect them to give us a qualified opinion in an area outside of their expertise. So, in an instance where we want the best qualified opinion we can get, the opinion of anyone who isn't an expert doesn't matter.

Yeah, there's some opinions that are so poor that they aren't worth entertaining. When's the last time you even heard of the Westborough Baptist Church? They're still doing their bs. They're just being driven into obscurity by the sheer terribleness of their opinions. Their opinions really don't matter. At best, they've turned into an obscure case study of what not to be.

There's a common sentiment that you are "entitled to your opinion" and everyone deserves a platform. I disagree with this sentiment because critical thinking and logic is not something that is intuitive to every person, these are skills that are learned.

If a population isn't granted the necessary philosophical tools to critically examine information and make a proper judgement, then they aren't prepared to discern between good and bad information. They don't know when they are getting tricked by having their biases catered to. IMO, this is why grifters and MLMs are so successful in the US; many Americans aren't educated and they are raised with American ideology that is not consistent with reality.

As such, unless you can formulate an opinion that's strongly based in established facts and logic (not like the right-winger meme, but actual studies to back up an argument), then your opinion should probably not be taken seriously.