You should define day.name
as a factor variable and use breaks()
to make the x axis readable.
Sample code:
library(ggplot2)
DelaysTimeofDayTotal$day.name= factor(DelaysTimeofDayTotal$day.name, levels= c("Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday", "Sunday"))
ggplot(DelaysTimeofDayTotal, aes(day.name, CRSDepTime)) +
geom_point(aes(y = DepDelay, color = "DepDelay"))+
geom_point(aes(y=ArrDelay, color = "ArrDelay")) +
scale_x_discrete(breaks=c("Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday", "Sunday")) +
labs(x= "Time of Day", y = "Delays in minutes", color="Delay Status") +
ggtitle(" Year 2005 - 2007 Average Delay for Time of Day") +
theme(plot.title = element_text(face="bold"))+
theme_minimal()
Plot:

Also, I would use rather geom_jitter()
to visualise the overlapping points.
Sample code:
ggplot(DelaysTimeofDayTotal, aes(day.name, CRSDepTime)) +
geom_jitter(aes(y = DepDelay, color = "DepDelay"))+
geom_jitter(aes(y=ArrDelay, color = "ArrDelay")) +
scale_x_discrete(breaks=c("Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday", "Sunday")) +
labs(x= "Time of Day", y = "Delays in minutes", color="Delay Satus") +
ggtitle(" Year 2005 - 2007 Average Delay for Time of Day") +
theme(plot.title = element_text(face="bold"))+
theme_minimal()
Plot:

Sample data:
DelaysTimeofDayTotal<-structure(list(CRSDepTime = c(510, 511, 512, 514, 515, 516, 517,
518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530
), DepDelay = c(-0.4529247, 4.4774775, -2.5608586, 1.9598214,
-0.32222292, -3.86666667, 4.91714286, -4.1439271, 6.2290914,
0.9113794, 2.4074074, 0.1282051, 1.1053494, 0.8019896, 0.69339187,
-2, 2.1190476, 2.74566427, 4.2522819, 1.0220646), ArrDelay = c(-1.34935901,
1.14414414, -0.67111557, -1.62053571, -2.6085016, -11.8666667,
9.4, -4.131957, 7.10905807, -0.57737008, 0.38888889, 1.10256441,
10.7990334, -5.47912288, -0.75862405, 8, -9.6190476, 4.943389978,
7.19421907, -0.88228056), day.name = structure(c(1L, 2L, 3L,
4L, 5L, 6L, 7L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 1L, 2L, 3L, 4L, 5L,
6L), .Label = c("Monday", "Tuesday", "Wednesday", "Thursday",
"Friday", "Saturday", "Sunday"), class = "factor")), spec = structure(list(
cols = list(CRSDepTime = structure(list(), class = c("collector_double",
"collector")), DepDelay = structure(list(), class = c("collector_double",
"collector")), ArrDelay = structure(list(), class = c("collector_double",
"collector")), day.name = structure(list(), class = c("collector_character",
"collector"))), default = structure(list(), class = c("collector_guess",
"collector")), delim = ","), class = "col_spec"), problems = <pointer: 0x1ac39d58>, row.names = c(NA,
-20L), class = c("spec_tbl_df", "tbl_df", "tbl", "data.frame"
))