Is it okay to work in a very old research area (in TCS)? I am a Ph.D. student works in theoretical computer science, I work on the area which is related to rings and my work is to design algorithms for problems related to rings. I did my bachelor in computer science not in mathematics.

I have a strong interest in this area, despite the fact that I have been told and suggested by multiple people to not pursue research in this field; I did it because of my interest. Due to my strange choice, I did not get much collaboration. I have been able to publish just a couple of research papers (not in the top tier) and will graduate after two semesters. I worked very hard, but the output I got is not even a fraction of what I have invested. 

I still like it, but the problems is that job opportunities, collaboration, and funds are very scarce. There are funds but only for those who publish in the top places. I am trying to publish in top places, but till now I was not successful. Many people around me have told me that my work is average. Some senior researchers advice me to be patient as it takes time to mature in this area.

Keeping the future in my mind, I see myself on a very different track as compared to others. My goal is to continue the research as well as get in some decent institute after my Ph.D. 

My question is it possible to be a successful researcher in this kind of area? What does it mean to do some significant research? Is it okay to work in a very old research area?
You have discovered a fundamental trade-off. If you work in an obscure field and do something significant it may be harder to get recognized, but you have the field to yourself and a few colleagues. But if you work in a popular area you might just get lost in the crowd and even get scooped in your latest findings. Neither is ideal. But either can be a path to success. 

However, you need a base from which to work. But so does everyone. In looking for an academic job you need to match yourself to requirements and also show that you have the potential to stand out, at least a bit. If you are competing with 40 other people just like you in background and research interests it will be difficult to stand out. But if you are unique you will raise questions about how you will fit in. It is, however, your case to make. 

It is easier, of course to do what you want when you are already well established (tenured). But you have to do what it takes to get tenured. But I don't think you need to defer doing what you like. You just need to do it well. 

Personally, I find your interests intriguing. But that is because I studied math but worked in CS. I'm certainly not unique in that. Being able to collaborate across departments might be seen as a plus in some institutions. So might joint appointments between two departments. Everything is a tradeoff. 

But spending your working life on things that don't much interest you just because they are "hot" seems like a waste of talent. Since you don't seem to be one of those fighting to get into one of the top ten institutes in the world, but can, perhaps, have somewhat more modest goals (top 50?, 100?) I think you can do fine. 

So, yes, you can be a successful researcher, but you will have to make that success yourself.