RNA processing in eukaryotes
As others have mentioned poly-A seems to be favoured because of higher concentrations of ATP with respect to other NTPs, due to its ubiquitous role. There is no real proof for this like most "why" questions. However, the ubiquitous presence of ATP is itself an evidence for a random selection of ATP and a subsequent expansion of its role. There is no thermodynamic explanation for this. The selection is likely to have been random.

Polyuridylation is also known to happen in eukaryotic cells (reviewed by Munoz-Tello et al., 2015) and one of the well known cases is that of the let-7 pre-miRNA (Heo et al., 2008). Polyuridylation marks this pre-miRNA for degradation.