The reincarnation of a lama isn't chosen or elected. He or she is found.
Man or woman
Historically the Dalai Lama has always been a man, but it has been my understanding that there is no rule that the next Dalai Lama can't be a woman. In fact, according to this news article the Dalai Lama once said that
If a woman reveals herself as more useful the lama could very well be reincarnated in this form.
This is confirmed in this interview
where the Dalai Lama:
... also said half-jokingly that if the Dalai Lama’s reincarnation is female, she must be very attractive. The reason is so that she will have more influence on others. If she is an ugly female, she won’t be very effective, will she?
In recent years, the Dalai Lama has also mentioned that he might be the last one. According to this BBC news article:
Whether another Dalai Lama came after him would depend on the circumstances after his death and was "up to the Tibetan people"
"The Dalai Lama institution will cease one day. These man-made institutions will cease," the Dalai Lama told the BBC.
These remarks were most likely politically motivated, because China very much wants to find the next Dalai Lama so they can 'control' him/her, just like they did with the Panchen Lama.
Finding a reincarnation
Suppose the Dalai Lama decides to come back after all. When a great Bodhisattva dies he1 can control the time and place of his rebirth. This is called the Tulku system and it makes sure that knowledge from a certain lineage is passed on. After his death a committee of high-ranked lamas of the same tradition as the deceased will gather with the task to find the reincarnation. This process can take several years and the committee will generally use a variety of sources.
In the case of the Dalai Lama it's customary to start with consulting the Nechung Oracle.
If the Dalai Lama is cremated the direction in which the smoke goes is also an important sign. It's possible for a prominent lama to have a vision or dream that reveals important details that help in finding him (e.g. certain characteristics of the location). Sometimes the deceased has left clues (e.g. a letter or specific personal objects) as to where he will be reborn.
Once a reincarnation is found, he will be tested to confirm that the right person has been found. For example, the 14th Dalai Lama was found at age 2 and one of his tests was to identify which items belonged to his predecessor.
If you are interested in this, there are a few films and documentaries that show how finding a reincarnation works. For example the (somewhat) popular movie Kundun starts with this IIRC, and I can recommend the documentary Unmistaken child, but that one may be harder to find.
1 for ease of reference I will use the words 'he/him' from here on, but as I mentioned earlier it can be a she/her.