How can I summarize by case insensitive?

Depending on the kind of aggregation you're doing, it may be useful to first summarize by name and then summarize again by tolower(name), so that your query converts significantly fewer strings to lowercase.

Doing so is possible, for example, in the following case, with a count() aggregation:

datatable(s:string)
[
    "abc",
    "def",
    "ABC",
    "AbC",
    "def",
    "ABc"
]
| summarize c = count() by s
| summarize c = sum(c) by s = tolower(s)