Dataframes in a list; adding a new variable with name of dataframe

I have a list of dataframes which I eventually want to merge while maintaining a record of their original dataframe name or list index. This will allow me to subset etc across all the rows. To accomplish this I would like to add a new variable 'id' to every dataframe, which contains the name/index of the dataframe it belongs to.

Edit: "In my real code the dataframe variables are created from reading multiple files using the following code, so I don't have actual names only those in the 'files.to.read' list which I'm unsure if they will align with the dataframe order:

mylist <- llply(files.to.read, read.csv)

A few methods have been highlighted in several posts: Working-with-dataframes-in-a-list-drop-variables-add-new-ones and Using-lapply-with-changing-arguments

I have tried two similar methods, the first using the index list:

df1 <- data.frame(x=c(1:5),y=c(11:15))
df2 <- data.frame(x=c(1:5),y=c(11:15))
mylist <- list(df1,df2)

# Adds a new coloumn 'id' with a value of 5 to every row in every dataframe.
# I WANT to change the value based on the list index.
mylist1 <- lapply(mylist, 
    function(x){
        x$id <- 5
        return (x)
    }
)
#Example of what I WANT, instead of '5'.
#> mylist1
#[[1]]
  #x  y id
#1 1 11  1
#2 2 12  1
#3 3 13  1
#4 4 14  1
#5 5 15  1
#
#[[2]]
  #x  y id
#1 1 11  2
#2 2 12  2
#3 3 13  2
#4 4 14  2
#5 5 15  2

The second attempts to pass the names() of the list.

# I WANT it to add a new coloumn 'id' with the name of the respective dataframe
# to every row in every dataframe.
mylist2 <- lapply(names(mylist), 
    function(x){
        portfolio.results[[x]]$id <- "dataframe name here"
        return