The use of the eye would appear to predate Islam (thinking of Eye of Horus), is it even older than the Egyptians? And how did it survive the rise of Islam? Were there any competing symbols too?
Why an 'eye' in the first place?
The evil eye - ayn al-ḥasūd in Arabic عين الحسود, chashm zakhm in Persian چشم زخم and kem gözler meaning evil eyeS in Turkish - is a motif borrowed from antiquity. It was reference in Ancient Greece and Rome (Plutarch, Herodotus, ) as well as Egypt, where there was a Great Goddess Eye, besides the Eye of Horus which you rightly mention. Isis also killed the King of Byblus just by looking at him. I'm not sure if there were any evil eyes before Egypt, but ancient India (Shahti and Shiva) also developed an 'evil eye' culture.
How did it survive Islam?
The evil eye actually exists in the Qu'ran. The Prophet Muhammad said:
"The influence of an evil eye is a fact..." (Sahih Bukhari, Hadith 5299)
"The impact of this evil eye is like a surge of fire that reaches a body. In bathing is the treatment to this fire surge … water will exhaust this surge of fire and its impact will be annulled.” (Fath al-Bari 16/268)
Some people in the Arab world say "Masha'Allah" when a compliment is made to ward off the evil eye, which in Arab culture is an eye of envy. An equivalent expression in Judaism is b'li ayin hara, or 'without the evil eye'. Blue eye talismans like the 'nazar' are also used to ward against the evil eye. Interesting, the Ancient Romans used a phallic talisman instead of a eye shaped one.
Al Bukhaari:
There is nothing wrong with taking precautions against the evil eye before it happens, and this does not contradict the idea of tawakkul (putting one's trust in Allah). In fact this is tawakkul, because tawakkul means putting one’s trust in Allah whilst also implementing the means that have been permitted or enjoined. The Prophet (peace and blessings of Allah be upon him) used to seek refuge for al-Hasan and al-Husayn and say: U’eedhukuma bi kalimat Allah al-tammati min kulli shaytanin wa hammah wa min kulli ‘aynin lammah (I seek refuge for you both in the perfect words of Allah, from every devil and every poisonous reptile, and from every evil eye).
BUT there are also writings AGAINST such superstitions from Islamic writers:
The Apostle of Allah said: You may live for a long time after I am gone, Ruwayfi', so, tell people that if anyone ties his beard or wears round his neck a string to ward off the evil eye, or cleanses himself with animal dung or bone, Muhammad has nothing to do with him.
Were there any competing symbols?
Firstly, the evil eye is blue because there was a folk belief in the Middle East that those with blue/green eyes can curse others. This is thought to be because such eye colours are rare in the region. (Daniels, et al., eds, Encyclopædia of Superstitions, Folklore, and the Occult Sciences of the World (Volume III)). Also, blue is traditionally associated with the sky, protection, and hence has a 'divine' or 'celestial' connotation.
As for competing symbols, the hand seems to have been a common talisman. This is known as al-kaff (the Hand of Fatima) and can be used to get rid of the evil eye.