Question: What does hidāyah mean?
It has such senses: to enter the true path after seeing right as right and wrong as wrong; to guide one to the true path; to turn away from dalālah [deviation, going astray] and false paths; to have īmān and to become a Muslim; a guide; the Qurān; tawhīd.
Hidāyah means guiding a person to the true path; being on the path Allahu taālā is pleased with; Allahs purging ones heart from all the gloom, distress and supplying it with comfort and expansion instead, thus granting complete easiness in observing His commandments and prohibitions; His subjugating a born slaves contentment to His qadā and qadar. As for the meaning of ihtidā, it is to attain hidāyah, that is, to become a Muslim, to choose Islam as ones religion.
The parts added in square brackets to the purport of undermentioned verses are explanatory statements from tafsīr books:
(Our Lord is He Who grants a distinctive feature to everything and then gives hidāyah [Who guides towards the true path].) [Sūrat-u Tāhā, 50]
(We made them all [Ibrāhīm, Ishāq, Yaqūb] imāms [guides] who would show hidāyah [the true path, Islam] by Our command [revelation]. We revealed to them the doing of good deeds, the performance of namāz correctly, the giving of zakāt. They were the persons who worshipped Us [they were not idolaters].) [Sūrat-ul-Anbiyā, 73]
(Allah guides whom He wills to the right path.) [Sūrat-ul-Baqara, 213]
(Those who accepted hidāyah [those who became Muslims by having īmān], Allah increased their hidāyah [their success in the true path, their obeying Islam] and inspired [explained] them the remedies for bewaring of evil.) [Sūrat-u Muhammad, 17]
(Whoever believes in Allah, Allah guides his heart to hidāyah [truth, Islam].) [Sūrat-ut-Taghābun, 11]
(The dwellers of Jannah, beneath which rivers flow, say, Praise be to Allah Who has made us attain to this with His hidāyah [by making us Muslims]. Had Allahu taālā not given us hidāyah [had He not made us Muslims], we could not have found this path on our own.) [Sūrat-ul-Arāf, 43]
(Those who believe and do pious deeds; because of their belief, their Lord guides them to Jannats [puts them in Jannats], beneath which rivers flow and which abound with blessings.) [Sūrat-u Yūnus, 9]
(O My Messenger, say: Whoever is an enemy to Gabriel is an enemy to Allah because he, by Allahs leave, has brought down the Qurān to your heart as a confirmer of the books sent before it, as a hidāyah [guide], and as a bearer of glad tidings.) [Sūrat-ul-Baqara, 97]
(When we listened to the hidāyah [the Qurān], we believed in it.) [Sūrat-ul-Jinn, 13]
(Allah does not give hidāyah [does not guide to the true path] to those who commit zulm [to those who take disbelievers for their friends and thus who do injustice to themselves].) [Sūrat-ul-Māida, 51]
(He leaves in deviation whom He wills, and He guides to hidāyah [the true path, Islam] whom He wills.) [Sūrat-ul-Fātir, 8]
(Allah guides to hidāyah [guides to Islam] whom He wills; He leaves in dalālah whom He wills.) [Sūrat-u Ibrāhīm, 4]
(We have sent down this Book to you so that you explain the controversial matters to them, and as a hidāyah [a guide showing the true path] and mercy for those who believe.) [Sūrat-un-Nahl, 64]
(Those who disbelieve the liqā-Allah [meeting Allah] and thus who stay away from hidāyah [from Islam, which is the true path] will certainly suffer the greatest loss.) [Sūrat-u Yūnus, 45]
(Hidāyah is only Allahs hidāyah [the true path is only Allahs path].) [Sūrat-ul-Baqara, 120]
(Those are the ones to whom Allah has given hidāyah [whom Allah has made attain to Islam].) [Sūrat-uz-Zumar, 18]
(Allah increases the hidāyah [their devotion to Islam] of those who have attained it [who have īmān and become Muslims].) [Sūrat-u Maryam, 76; Sūrat-u Muhammad, 17]
(He makes them attain to hidāyah [He makes them attain to īmān].) [Sūrat-u Muhammad, 5]
(They bought dalālah instead of hidāyah [the true path].) [Sūrat-ul-Baqara, 175]
(Allah has sent His Messenger with Islam, which is hidāyah and the true religion. He has made Islam superior over other religions. Allah suffices as a witness that He is a Prophet [that Muhammad alaihis-salām is the True Prophet].) [Sūrat-ul-Fath, 28]