Let's say you want to keep all the rows where Name == "A" and the columns Name and Value. I would use code like the following.
# Your code
library(readxl)
setwd("~/Documents/Kent/Students/FYP/2020-21/SRE_loneliness/data/OlderAdults/data")
file.list <- list.files(pattern='*.xlsx')
df.list <- lapply(file.list, read_excel)
#Added code
library(dplyr)
library(purrr)
names(df.list) <- file.list
FilterFunc <- function(DF) filter(DF, Name == "A") |> select(Name, Value)
map(df.list, FilterFunc) |> bind_rows(.id = "File")