I presume Edenberry is a mistranscription of Edenderry, which is one of the larger towns in the county. (There's another town named Edenderry in Co. Antrim)
Do you know what denomination your Muldoon family were ?
There are some transcripts of parish records for Co. Offaly on the pay-website at RootsIreland, including some for Edenderry. The free index is showing a baptism of two Bridget Muldoons in the county between 1852-1862, but neither are in Edenderry RC or CofI parishes, or show a match for the parents names.
see :
Introduction to Irish RecordsShane