Should I be an author on another PhD student's paper if I went to their meetings and gave advice?
Actually, I think you should relax and take your advisor's advice. Collaboration is a good thing, and it is a two-way street. You give a bit and you get a bit. I assume you got an acknowledgement in the paper for your help. I don't think it would be appropriate if you weren't. But authorship is a different thing.
You contributed ideas. Research seminars are often organized to give ideas to researchers but the members don't become co-authors in the normal case.
Congratulate your colleague and, as your advisor suggests, spend your effort on your own work, not raising an objection to someone else's.
But, it is good that you contributed ideas. Do that a lot and you will have a lot of people willing and happy to work with you. Occasionally you may need that help.
Pay it forward.
I am planing in taking it to the head of department.
Let me assure you that this is a bad idea. 99 times out of 100 the department head will not intervene in these matters. Moreover even if they do (again, super unlikely), and you get things your way with this paper, I assure you that this will forever mar your relationship with your advisor. I would be extremely upset if one of my students went over my head like this.
How about you have a discussion with your advisor about how you feel? What constitutes author worthy contribution highly varies between research groups, so I would not be so quick to decide that your contribution suffices (nor am I in a better position than your advisor to make this call). Maybe you can be more involved in follow ups? In relating this work to your own? As Buffy mentions, you’re not just letting it go because that’s the way it is, it’s also because being adversarial will have far reaching repercussions beyond this one paper!
In my experience, a collaborative approach pays dividends in the long run: Be the person people want to talk research with!
EDIT: To clarify: It is impossible to judge whether X number of hours warrants coauthorship status, as this is very discipline/relationship related. Some PIs think that even a short conversation about the paper and some suggestions warrant coauthorship, others think that unless you actively participate in writing you shouldn't be a coauthor.
Unless something outright unethical is happening (e.g. the OP made major contributions, which is not obvious from the OP), I don't think they have a case, certainly not one department head I've met would intervene in..
You look like you are an expert on a field that your supervisor isn't, and that expertise is necessary for a collaborator to make progress on his thesis. Therefore, I would expect that your supervisor would motivate both of you to work together, so that the other PhD student can benefit from your expertise, and you could benefit from working on a related subject you know about. In that case, co-authorship would be reasonable and come of naturally. I also believe that you would generously offer co-authorship to your collaborator, if things had happened the other way around (i.e. he had provided advice the way you have).
But things evolved differently, so:
Do you deserve to be a co-author?
In my opinion, yes. Judging from your descriptions, it looks like you've been doing lots of "supervisor" work here. As others have noted, a supervisor is anyway included regardless of their contribution. To my understanding, and based on my academic experience as a PhD student, this is not just a "typical" practice; since the supervisor is expected to get co-authorship, they should provide actual supervision (advice, intuition etc). Providing it through a delegate is fine, but I would expect that the delegate receives proper credit.
Should you take action?
No. Unless you don't mind hurting your relationship with your PhD supervisor and possibly triggering a conflict with him. Given the circumstances, I'd suggest you give up on co-authorship, but express your feelings to your supervisor.
Is this fair?
No, at least in the way I perceive academic ethics. Unfortunately, ethics and rules in academia are easily violated in subtle ways. If you are not the one holding power, there's very little you can do without risking a conflict with people who have huge control over your academic future.