What can I do to convince my advisor of my ideas?

First step is to have good ideas. :)

But wait, you are starting your phd, by definition, you wouldn't know what a good idea is, you do not have the experience (usually, let's leave the genius outliers aside for now).

When I have a crazy idea, I develop it a bit, to see where it leads. If leads somewhere interesting, I prepare a nice, informative presentation, and schedule a meeting with the boss. Sometimes I get shot down, sometimes the idea is accepted, but "preparing the field" is important. You need to present your ideas properly, in a well prepared manner, to increase the chances he would understand it. And clearly presenting ideas is the bread and butter of research.

With that said, have you tried an honest conversation with him on this aspect? Nothing wrong with an student telling me he is lost and he needs a concrete direction. Even better if that direction has specific, detailed objectives, including some easily attainable ones to keep the morale high.

Be honest, be polite, be patient, and be clear.

Ideas alone will not be sufficient in some cases. You need to provide supporting evidence and if possible, proof.

For example, let's say you are analyzing the data emerged out of an experiment and find something that doesn't match with the expected outcome. If you come up with an idea in your mind which you think might explain the phenomena, don't just jump up from your seat and run to your adviser, rather you must check and verify the details with care. You may follow the usual research methodology like a background check for the published articles at first, eliminating the obvious experimental pitfalls, errors in analysis and so on. If your idea still stands, then you may create a detailed report with everything you have done to prove your claim(a LaTex PDF?) and submit it to your supervisor. From my experience this is an effective way to communicate novel ideas.

I think the most practical things to do is to ask your advisor for a starting point.

1. Ask your advisor for specific books, articles to read.

2. Work through these books/articles (and their exercises/problems) and ask questions where and when you can (without annoying your advisor, of course).

3. Also tell your advisor what you're planning on doing next. This way you know if you're drifting too far from your advisor's expertise, or into territory that your advisor deems uninteresting. (Knowledge that your advisor can't verify or doesn't find interesting is unlikely to play a big role in your thesis.)

4. Ask if problems/questions you come across can be asked in a different (or more general) context. If the answer is

   • "Yes of course!", do it as an exercise (and show your advisor afterwards)
   • "No!", find a counter-example
   • "I don't know [but it doesn't matter]", leave it
   • "I don't know [but it would be interesting]", try to find out more.

Problems from this last category are often the ones that make it into your thesis, but as you see, you don't get there without initial (specific) input from your advisor.