Final answer:
5'-Capping is the post-transcriptional modification that enhances mRNA stability, facilitates ribosome binding, and prevents exonuclease degradation at the 5' end by adding a protective 7-methylguanosine cap.
Step-by-step explanation:
The type of eukaryotic post-transcriptional modification that enhances stability, is important for ribosome binding, and prevents degradation by exonucleases at one end is 5'-Capping.
During the transcription process, while the pre-mRNA is still being synthesized, a 7-methylguanosine cap is added to the 5' end of the growing transcript via a phosphate linkage. This cap acts as a protective shield against exonuclease degradation and is recognized by factors involved in protein synthesis, which then help to initiate translation by ribosomes.