Will a Muslim be held accountable for his intention in naming his sons "Muhammad" and "Abdullah" so that they may fulfill one of the conditions for being the awaited Mahdi, knowing that he is keen on raising them well and strives hard in that regard?

It has not been proven that the Mahdi is specifically from the progeny of Hussein. What is established is that he is from the Household of the Prophet (Ahl al-Bayt). More than one scholar has expressed the view that he will be from the progeny of Hasan, based on a hadith narrated by Abu Dawud from Ali (may Allah be pleased with him), and what Al-Suyuti quoted from Al-Qari, Ibn Taymiyyah, and Ibn al-Qayyim, who indicated that Hasan abandoned the Caliphate for the sake of Allah, so Allah destined from his offspring one who would establish the true Caliphate.