Lower p-values apparently give you more confidence in the alternative hypothesis

"But an isolated finding, especially when embodied in a 2 X 2 design, at the .05 level or even the .01 level was frequently judged not sufficiently impressive to warrant archival publication." (p. 554)

From: Melton, A. W. (1962). Editorial. Journal of Experimental Psychology, 64, 553–557.

According to Gigerenzer et al (Published in: D. Kaplan (Ed.). (2004). The Sage handbook of quantitative methodology for the social sciences (pp. 391–408)), this quote is where the common convention comes from to use p-values as a measure of one's belief in a result.

Gigerenzer et al write: 

"Editors of major journals such as A. W. Melton (1962) made null hypothesis testing a necessary

condition for the acceptance of papers and made small p-values the hallmark of excellent

experimentation."