I feel this is a difficult situation, with a number of variables that should influence your decision:
- Is your advisor completely gone (e.g., in complete retirement, gone to industry, or even dead), or has he just left for a different academic post? If he is completely gone, you may need to find somebody to supervise you in your day-to-day research. If he has just moved on to a different university, it is possible that you want somebody to "formally" be your PhD advisor, but practically keep on working remotely with your old advisor. Whether you even want to keep on working remotely with your old advisor depends on how advanced you are and how well you worked with her/him beforehand.
- Are you advanced enough that you can finish your thesis more or less on your own? That is, do you need an advisor or an examiner for your thesis? If it is the latter, then pretty much anybody who is sufficiently well-versed in your area could act as your advisor for the finishing sprint. If you actually feel like you still need a lot of input to get your thesis ready, it will be harder to find somebody who can and is willing to take you on midway. If you are advanced and your advisor is only moving to a different university, it should be reasonably easy for you to find a "local proxy" without having to adapt your research plan too much.
- How set-in-stone is your research plan? Or, in other words, how ready are you to adapt to the research agenda of another advisor? Generally, if you really don't want to change your research, but you also feel like you still need a lot of input, it is likely that you will have problems finding a suitable new advisor. If you are happy to change research topics, finding an advisor should not be more difficult than the first time.
- How does your funding work? That is, do you "only" need a new advisor, or do you need a new advisor and a new funding source (e.g., project, grant)? Clearly, the former case will make it much easier to find a new advisor. In the second case, you will need to be willing to change your research plan somewhat, as few professors will be ready to dish out grant money if the student is not actually working on their projects.
- Are you a good student? Ok, this may be somewhat controversial, but clearly if you have been an overperformer so far, it will be substantially easier to convince another professor to take you on. Similarly, if you already have great publications in your research, it will be much easier to convince a professor that you want to continue working on this concrete research idea and not change to whatever other project your new advisor has in mind.
- Are you willing to change universities? If so, maybe coming along with your old advisor is an option?
So how does this help you? In a first step, you need to reflect on all of these points to figure out what kind of advisor you actually want/need. Then go over the list of potential faculty that formally could serve as advisor. Think about how your collaboration with each of those potential advisors could work, and what you would need from them. Get in touch with the ones that seem most suitable (as you already study there I am assuming you can just send them a mail, or have your current advisor put you in contact), and discuss the issue with them. Select the one that (a) is willing to take you on, and (b) you feel is most consistent with your answers to the questions above.
