Interpreting lagged IVs in simple linear regression? If we have dependent variable y and independent variable x (where x and y are measured in the same month for each data pair, and represent month over month financial returns), running a simple linear regression tells us what happens when x moves by, say, 1%. If we assume a regression coefficient of 2, this means for every 1% increase in x, y will increase by 2%.
Implied above, assume x and y are both time series data, measured monthly. How does the above interpretation change if we run a regression between x measured in month i and y in month i+1?

The interpretation is very similar. The coefficient on lagged $x$ tells you the change in $y$ at time $t$ (in units of $y$) associated with 1 one unit increase in $x$ at time $t-1$ ($x$ in the previous month).
For example, the number of people applying for unemployment in September can be modeled using the number of people who lost their jobs in August.
There are two issues with your interpretation.
The percent interpretation is not correct unless you transformed the data to achieve that (say by logging both $x$ and $y$) and the coefficient is small (otherwise, the approximation is not so good).
If your data is in percent already, then the relationship is in percentage points, nor percent.
The causal interpretation of $x$ changing $y$ is also not correct unless you make some strong assumptions. For example, this month's expected income may be higher for those who drove a Mercedes last month, but that's not the primary causal relationship.
