Below is a code that renames the column names of each data.frame in a list in a way so that names of data.frames are added to original column names.

# example data
a <- data.frame(col1 = 1:10, col2 = 10:1)
b <- data.frame(col_01 = 11:20, col_02 = 20:11)

# list of data.frames
list_of_df <- list(a, b)

# names of data.frames
names(list_of_df) <- c("a", "b")

# my sequence and names of data.frames in a list
my_seq <- seq_along(list_of_df)
my_list_names <- names(list_of_df)

# procedure
for (i in my_seq) {

  names(list_of_df[[my_seq[i]]]) <- 
    paste(my_list_names[i], names(list_of_df[[my_seq[i]]]), sep = "_")

}

list_of_df

$a
   a_col1 a_col2
1       1     10
2       2      9
3       3      8
4       4      7
5       5      6
6       6      5
7       7      4
8       8      3
9       9      2
10     10      1

$b
   b_col_01 b_col_02
1        11       20
2        12       19
3        13       18
4        14       17
5        15       16
6        16       15
7        17       14
8        18       13
9        19       12
10       20       11