OK, here's a proposed solution I came up with. Start with the overall open rate for all emails regardless of time of the week. Use that number, and your intuition for how much variation you are likely to see between different days and times (perhaps informed by studies on this subject that people have already done) to construct some prior distribution over the open probabilities you think you're likely to see. You'll want to choose a distribution over the interval (0, 1) only... I'm not sure if this one or this one is better in this particular case. Then for each hour of the week, use maximum-a-posteriori estimation (this seems like a brief & good explanation) to determine the mode of the posterior distribution, after you've updated on all of the open data you've observed. (This provides an explanation of how to do this.) The mode of an hour's distribution is your probability estimate that an email sent during that particular hour of the week will be opened.

Given those probability estimates, you can figure out how many opens you'd get if emails were allocated optimally throughout the week vs how many opens you'd get if they were allocated randomly and figure out if optimal allocation would be worthwhile to set up.