purrr::map2() is known to not function well with dplyr::rename(). Defining lambda in call to map2() solves the problem.https://github.com/tidyverse/purrr/issues/541

rename_nested_tibble_column.R

library(tidyverse)

data <- tibble::tribble(
  ~phase,      ~fc_dt,        ~oil,          ~q,        ~gas,  ~t,  ~mean,
  "water",  "4/1/2017", 4.602535714,     1.70775, 73.55432143,  1L, 10.60021233,
  "water",  "5/1/2017", 4.414064516, 5.845258065, 48.88796774,  2L,  7.86293583,
  "water",  "6/1/2017", 3.215866667, 4.149533333, 43.78976667,  3L,  6.47610122
)

nest_data <- data %>% nest(-phase)

nest_data

expected_output <- nest_data %>%
  mutate(test = map2(data, phase, function(x, y) {
    names(x) <- str_replace(names(x), "^mean$", y)
    x
  })) 

expected_output$test

 library(tidyverse)
  data <- tibble::tribble(
    ~phase,      ~fc_dt,        ~oil,          ~q,        ~gas,  ~t,  ~mean,
    "water",  "4/1/2017", 4.602535714,     1.70775, 73.55432143,  1L, 10.60021233,
    "water",  "5/1/2017", 4.414064516, 5.845258065, 48.88796774,  2L,  7.86293583,
    "water",  "6/1/2017", 3.215866667, 4.149533333, 43.78976667,  3L,  6.47610122
  )
  
  data %>% 
    nest(-phase) -> nest_data
  
  # fails
  nest_data %>%
    mutate(test = map2(data, phase, ~ rename(.x, !!.y := mean)))
  
  # standalone works
  map2(nest_data$data, nest_data$phase, ~ rename(.x, !!.y := mean))
  
  # workaround works
  nested2 <- nest_data %>%
    mutate(test = map2(data, phase, ~ rename(.x, !!expr(phase) := mean))) %>% 
    unnest(test)