I'm director of graduate admissions for the math Ph.D. program at a research I department. We are certainly interested in assessing research potential. But we don't judge this by published papers; almost no undergrad has any, and when they do, the actual material they contain is usually not of great interest.
It could certainly be helpful to start exploring research. You should do this under the supervision of a professor (whether at your own institution in the context of a thesis or capstone project, or at an REU) who will be able to attest, in a recommendation letter, that you are very likely to prove interesting theorems in the future.
This is a question of interest to most mathematicians who are research active and not slowly but surely knocking off important problems in their field at the rate of one per paper. (I think I could have ended the previous sentence at the word "active" without much affecting the meaning!)
I think the answer is ultimately quite personal: you are free to set your own standards as to how much of your work to publish. I myself understand the psychology both ways: on the one hand, math is usually long, hard work and when you finish off something, you want to record that accomplishment and receive some kind of "credit" for it. On the other hand, we want to display the best of what we have done, not the entirety. This position is well understood in the artistic and literary world: e.g. some authors spend years on works that they deem not ready to be released. Sometimes they literally destroy or throw away their work, and when they don't, their executors are faced with difficult ethical issues. (This is roaming off-topic, but I highly recommend Milan Kundera's book-length essay Testaments Betrayed, especially the part where he details the history of how after Kafka's death, his close friend Max Brod disobeyed Kafka's instructions and published a large amount of work that Kafka had specifically requested be destroyed. If Brod had done what he was told to do, the greater part of Kafka's Oeuvres -- e.g. The Trial, The Castle, Amerika -- would simply not exist to us. What does Kundera think of Brod's decision? He condemns it in the strongest possible terms!)
Another consideration is that publication of work is an effort in and of itself, to the extent that I would not say that anyone has a duty to do so, even after releasing it in some preprint form, as on the arxiv. A substandard work can be especially hard to publish in a "reasonable" journal. I have a friend who wrote a short note outlining the beginnings of a possible approach to a famous conjecture. She has high standards as to which journals are "reasonable", and rather than compromise much on this she determinedly resubmitted her paper time after time. And it worked -- eventually it got published somewhere pretty good, but I think she had four rejections first. I myself would probably not have the fortitude to resubmit a paper time after time to journals of roughly similar quality.
As you say, though, one advantage of formal publication is that the paper gets formal refereeing. Of course, the quality of this varies among journals, editors, referees and fields, but speaking as a number theorist / arithmetic geometer, most of my papers have gotten quite close readings (and required some revisions), to the extent that I have gained significant confidence in my work by going through this process. I have one paper -- my best paper, in fact! -- which I have rather mysteriously been unable to publish. It is nevertheless one of my most widely cited works, including by me (I have had little trouble publishing other, lesser papers which build on it), and it is a minor but nagging worry that a lot of people are using this work which has never received a referee's imprimatur. I will try again some day, but like I said, the battle takes something out of you.
Finally, you ask about how it looks for your career, which is a perfectly reasonable question to ask. I think young mathematicians might get the wrong idea: informal mathematical culture spends a lot of time sniping at people who publish "too many papers", especially those which seem similar to each other or are of uneven quality. Some wag (Rota?) once said that every mathematician judges herself by her best paper and judges every other mathematician by dividing his worst paper by the total number of papers he has published. But of course this is silly: we say this at dinner and over drinks, for whatever reasons (I think sour grapes must be a large part of it), but I have heard much, much less of this kind of talk when it comes to hiring and promotion discussions. On the contrary, very good mathematicians who have too few papers often get in a bit of trouble. As long as you are not "self plagiarizing" -- i.e., publishing the same results over and over again without admission -- I say that keeping an eye on the Least Publishable Unit is reasonable. Note that most journals also like shorter papers and sometimes themselves recommend splitting of content.
So, in summary, please do what you want! In your case, I see that you have on the order of ten other papers, so one more short paper which is in content not up there with your best work (I am going entirely on your description; I don't know enough about your area to judge the quality and haven't tried) is probably not going to make a big difference in your career. But it's not going to hurt it either: don't worry about that. So if in your heart you want this work to be published, go for it. If you can live without it, try that for a while and see how you feel later.
Best Answer
After writing a manuscript (which it seems you may have already), go through it and revise it a few times until you feel that it is in a polished form. Then you could ask your professors to read it and provide some feedback and revise accordingly. This revision/feedback process will be a good experience for practicing and getting a feel of what the process writing research mathematics is like. Plus, you may come across potential generalizations or other cases you may not have previously considered.
It's important to stress here that you should listen to your professors. They will generally know what journals / media that your paper would be suitable for. For instance, they could advise you on whether it would be worth doing any of the following:
These could definitely be valuable experiences to get a feel for mathematics research!
Even if you don't publish your current research, the experience is usually more useful in the following two ways than the actual mathematics:
Regarding (1), use this process to see if research mathematics is something that you want to pursue! If anything, you'll learn how to present logical arguments cohesively.
Regarding (2), explore the connections of what you are studying to fields of mathematics that you haven't learned yet. Perhaps when looking at generalizations or applications of your result, you'll find that you'll need some deeper mathematical theory X. Use this as motivation to go learn X! Maybe after studying X, you'll have a much better understanding of your previous results when going back to it. Who knows, maybe the ideas from your initial research can be useful for something else much further down the line, regardless of whether it was publishable in its initial form!