As you discovered with the answer and the request for arbitration, there will be the need for you to appoint an arbitrator, if it is a sole arbitrator you will need to agree with your counter party on this issue, and if you don’t agree, then it will be up to the judge by the request of one of the parties under an ad hoc arbitration, and if it is institutional arbitration this will be a task by the arbitral institution.
However, if it is a three member panel, then you are free to appoint your arbitrator, with this, you will need to make a conscious choice, you will perform what is called in the industry as a beauty contest, meaning that you will have interviews with possible arbitrators in order to be appointed to the dispute at hand.
But, what to expect and what to look for?
First, for my point of view make sure that your arbitrator actually speaks the language of the arbitration, as well as the possible second language of the contracts and documentation created, since it is of crucial importance that your arbitrator and the arbitral tribunal as a whole understands you.
Second, availability of time, and I cannot stress this enough, have an arbitrator that actually will devote her time to your dispute, that she will do her outmost diligent use of time in order to understand the dispute as well as to read the case files, you need this, and this is why you are appointing your arbitrator, also, make sure that your arbitrator will agree on exclusive confidentiality and not resort to junior lawyers for the reading the memorials, the evidence and drafting of the orders and the award.
Also, it is relevant that your arbitrator has previous experience in arbitration, whether as an actual arbitrator, secretary to the tribunal, part of the arbitral institutions, or if not, as a lawyer representing clients in arbitrations, with a good knowledge of the subject matter.
My personal ranking of what to look for an arbitrator is first the language, time availability, knowledge of the subject matter, and least, previous experience in arbitrations.
Why I rank in the last part the previous experience as an arbitrator, well, because even if international arbitration may appear as an intricate dispute resolution mechanism, it is a process that reflects what is commonly practice in any judicial arena where two parties have a legal dispute, and if you have a great lawyer with great knowledge of the law, then you are in a good standing on what to expect from your arbitrator.