How can I change the legend labels in R (ggplot2)

Personally, I would just add a column with the transformed values, i.e. instead of:

dat <- dat %>% mutate(Monthly_Gain = as.factor(Monthly_Gain))

I would use:

dat <- dat %>% mutate(`Monthly_Gain_%` = as.factor(Monthly_Gain * 100))

I would then use Monthly_Gain_% as my color variable.

dat %>% ggplot(aes(Desired_Income/1000, Account_Size/1000, color = `Monthly_Gain_%`)) +
    geom_point() +
    geom_line() +
    xlab("Desired Income in Thousand Dollars") +
    ylab("Required Account Size in Thousand Dollars") + 
    ggtitle("3% to 5% per month account growth") + 
    labs(col = "Monthly Gain") +
    theme(plot.title = element_text(hjust=0.5))

scale_color_manual() will also work, but may require more tinkering with the colors, depending on your needs. For example, to get:

You would load RColorBrewer and use:

library(RColorBrewer)

dat %>% ggplot(aes(Desired_Income/1000, Account_Size/1000, color = Monthly_Gain)) +
    geom_point() +
    geom_line() +
    xlab("Desired Income in Thousand Dollars") +
    ylab("Required Account Size in Thousand Dollars") + 
    ggtitle("3% to 5% per month account growth") + 
    labs(col = "Monthly Gain") +
    theme(plot.title = element_text(hjust=0.5)) + 
    scale_color_manual(labels = perc_title, values = brewer.pal(8, "Spectral"))

If you simply want to use the default colors as you have above, use scale_color_discrete() instead (scale_color_hue() would also work):

dat %>% ggplot(aes(Desired_Income/1000, Account_Size/1000, color = Monthly_Gain)) +
    geom_point() +
    geom_line() +
    xlab("Desired Income in Thousand Dollars") +
    ylab("Required Account Size in Thousand Dollars") + 
    ggtitle("3% to 5% per month account growth") + 
    labs(col = "Monthly Gain") +
    theme(plot.title = element_text(hjust=0.5)) + 
    scale_color_discrete(labels = perc_title)