Thanks for the post. If you use any of the major statistical software such as SAS, SPSS and STATA, you should be able to calculate the sampling error and the confidence interval of median using the complex survey module. In DHS, we use SAS. If you use SAS, you can use the codes:

PROC SURVEYMEANS median DATA=your data;
VAR XXX (name of your edited survival time variable);
WEIGHT YYY (name of your weight variable);
STRATA ZZZ (the stratification variable);
CLUSTER V001;
RUN;

It is worth to note that SAS calculates a true interpolated median, while SPSS and STATA calculate the median value of the variable, not an interpolated median.
Hope this helps.
Ruilin Ren