With respect to the meaning of Hujjah, it could be said that the word 'Hujjah' would be applied to the thing (matter) which we can infer or argue to it (as a brief description). (Read More... but source not in English)
The word "hujjah" has been used in different occasions in the holy Qur'an at verses like 4:165 and 8:149.
But in context to Shia Imams,
Hujjat means proof or demonstration. When used in reference to Imams, it is to point to their status as being living signs of Allah or His proofs. In Shia traditions, there are many references to Imams by the concept of 'Hujjah.' In fact, Muhammad Kulayni titled his fourth hadith collection about the subject of Divinely-appointed authorities and Imam al-Mahdi as the "Book of the Proof" or Kitab al-Hujja (Credits: infatuated here)
Shia Islam believe Hazrat Mahdi (Imam Zaman) is considered as the last Hojjah of Allah. That’s why he (Imam Zaman) is famous as Hojjat-ibn-al-Hasan. (Bahar-al-Anvar, Volume25, H6 and Volume97, Page 343.)
Davood ibn Qasem quoted I heard from the tenth Imam (Imam Ali al-Naqi (Hadi a.s.)) that was declaring:
«الخَلَفُ مِنْ بَعْدِی الحَسَنُ فَكَیفَ لَكُمْ بِالْخَلَفِ مِنْ بَعْدِ
الخَلَفِ فَقُلتُ لِمَ جَعَلَنِی اللّهُ فِداكَ فَقَالَ: اِنَّكُم لا
تَرَوْنَ شَخْصَهُ وَلایحِلُّ لَكُم ذِكْرُهُ بِاسمِهِ فَقُلتُ كَیفَ
نَذكُرُهُ فَقَال: قُولُوا الحُجَّةُ مِنْ آلِ مُحَمَّدٍ»؛(7)
My successor is my son, Hasan. How will be your feeling concerning the
successor of my successor? I said “God sacrifices me, why? Since you
won’t be able to see him …, He declared: you call him hojjat from the
household of the Muhammad (pbuh). / 7- كتاب الغیبة، ص 202، ح 169؛
كافی، ج 1، ص 328، ح 13
Furthermore, it (his name) has applied in majority of the narrations as “Al-Hojjat Men Ale Mohammad” «الحُجَّةُ مِنْ آلِ مُحَمَّدٍ») / 8- ر.ك: همان؛ شیخ صدوق، علل الشرایع، ج 1، ص 245؛ كمال الدین و تمام النعمة، ج 2، ص 648.