What is the best way to learn the names of the techniques in Aikido?

Each ryuha will have different names for the techniques. Whether it's the Aikikai's Ikkyo or the Tomiki style's Ikkajo, it is the same basic technique. But your ability to comunicate with your classmates will improve if you learn the local "dialect".

It is helpful (and traditional in many schools) for the instructor to name the techniques after demonstrating them to the class. Of course many students are not in a position to request changes to teaching methodology.

As with most things in Aikido, the best way is to approach your sempai and ask for assistance. When a technique is taught whose name you do not know, follow up with a sempai after class. Ask for the name, and maybe the chance to practice it a couple more times (both as uke and nage).

Good sempai will realize your desire to learn and improve and be happy to fill that roll.

After all that - finding a book written by an authority in your ryuha, using the same terminology that your dojo uses, is always helpful. Again, your sempai are a good resource: ask them what books they recommend.