A quick look at Stair na Gaeilge yields this (in Kim McCone’s chapter An tSean-Gaeilge agus a réamhstair — “Old Irish and its prehistory”)…
21.2 … It can be seen that use is made of the suffix *-(i)yā to make abstract nouns in IE itself (e.g., Gr. phil-ó-s ‘beloved’, phil-ía ’fondness’). The -e (MW -ed) that descended from it was a common way of forming feminine abstract nouns (13.5) from adjectives in Old Irish, e.g., dílis, díl†s-e; maith ‘good’, maithe ‘goodness’; gor ‘fervent’ (MW gwar) < *gwar-os (< *gwr̥-), goire ‘fervour’ (MW gwared) < *gwar-iyā (7.4). -tūt- was also available (17.1) and a couple of other suffixes were developed in (Insular?) Celtic itself to derive abstract nouns from other nouns, in particular *-axtā or *-yaxtā (OI -acht or -echt, MW -(i)aeth; [*-i-+] *-ak- + *-tā) and *-assu-s (OI -as, MW -as): e.g., OI marcach, MW marchawc ‘(horse-)rider’ and OI marcaigecht, MW marchogaeth ’horsemanship’, OI tigern(ae), MW teyrn ‘lord’ and OI tigern-as, MW teyrn-as ‘lordship’.
Kim McCone, Damian McManus, Cathal Ó hÁinle, Nicholas Williams, Liam Breatnach, eds. (1994). Stair na Gaeilge. II:21.2 “An tSean-Gaeilge agus a réamhstair: Gnéithe de dhíorthú na substainteach”, page 127.
My translation. OI = Old Irish, MW = Middle Welsh.
A connection to aigid seems unlikely to me.
Note that beocht may be more complicated than it looks. The older form is béodacht and there was also a form using the *-assu-s suffix mentioned above — béodas (the modern beos probably only exists in the dictionary).