Say a 5% bond using 30/360 convention, 2 coupons per year. Last coupon payment was on 2016-04-01. Now 2016-10-01 is weekend and the coupon is paid on 2016-10-03. Is this coupon 2.5 or slightly more than 2.5?
What is accrued interest on 2016-09-30? On 2016-10-03?
For the vast majority of bonds, as other commenters have pointed out, coupon sizes are generally not affected by bad days (i.e., holidays and weekends), so for a bond with semi-annual coupon payments, the coupon size will (almost) always be as simple as $c/2$. Some exceptions are:
Bonds with irregular first coupon periods: The first coupon period spans from the dated date (aka the first interest accrual date) and the first coupon date. If this period is not exactly a full coupon period (can be longer or shorter), then the coupon size must be calculated as $\text{DCF} \times c$, where $\text{DCF}$ is calculated as the day count fraction between the dated date and the first coupon date (usually NOT adjusted for holidays/weekends) using the correct day count convention.
Bonds with irregular last coupon periods: Similarly, the last coupon period, spanning from the penultimate coupon date and the maturity date, may not be a full coupon period. In these cases, the coupon size is also calculated as $\text{DCF} \times c$, where $\text{DCF}$ is calculated as the day count fraction between the penultimate coupon date and the maturity date (neither of which is bad-day adjusted, most of the time).
Bonds whose cashflows that follow the "exact accrual rule": These are quite rare and the only ones that come to mind are Thailand government bonds.** The cashflows for these bonds are based on the actual number of days for every coupon period.
As to accrued interest, it should always be calculated using the day count fraction between the previous coupon date (NOT bad-day adjusted) and the settlement date (bad-day adjusted).
For the example you cited, the coupon size should be 2.5, paid out on October 3. The accrued interests should be:
** Another exception is term CDs. Mayle (1993), a standard reference, notes that "The interest flows for a term CD differ from those of any other periodic security in that the amount of each flow is determined by the number of days in its period as opposed to the number of periods per year."
Doing the calculation on the FINRA calculator gives an estimate -
Assuming purchase date of 2016-04-01, 2016-09-30 would be a hold period of 179 days:
Accrued Interest 2.486 %
Value of Accrued Interest $24.86
And 2016-10-03 would be a holding period of 182 days:
Accrued Interest 2.528 %
Value of Accrued Interest $25.28
Granted, this does not account for the settlement period of the bond (which will have to be included when traded) which will raise your final accrued interest to 184 holding days:
