Proteins are among the most important constituents of biological systems. Because all protein-coding genes have a noncoding ancestral form, the properties of noncoding sequences and how they shape the birth of novel proteins may influence the structure and function of all proteins. Differences between the properties of young proteins and random expectations from noncoding sequences have previously been interpreted as the result of natural selection. However, interpreting such deviations requires a yet-unattained understanding of the raw material of de novo gene birth and its relation to novel functional proteins. We mathematically show that the average properties and selective filtering of the "junk" polypeptides of which this raw material is composed are not the only factors influencing the properties of novel functional proteins. We find that in some biological scenarios, they also depend on the variance of the properties of junk polypeptides and their correlation with the rate of allelic turnover, which may itself depend on mutational biases. This suggests for instance that any property of polypeptides that accelerates their exploration of the sequence space could be overrepresented in novel functional proteins, even if it has a limited effect on adaptive value. To exemplify the use of our general theoretical results, we build a simple model that predicts the mean length and mean intrinsic disorder of novel functional proteins from the genomic GC content and a single evolutionary parameter. This work provides a theoretical framework that can guide the prediction and interpretation of results when studying the de novo emergence of protein-coding genes.