電話のベルが聞こえたとき、無視しようかとも思った。
When I heard the phone's bell I wondered if I should ignore it.
We have 無視する = "to ignore",
無視しようか = "will I ignore it?"
無視しようかと思った = "I thought 'will I ignore it'".
I can't figure out what も is doing. Is it adding a sense of 'even'? As in 'I even thought that I should ignore it'?
Yes, it is the も meaning "also/even."
This も after quotative と is fairly common, and while it could certainly be the emphatic "even" sense (the larger context would help make it clear if we should be surprised by the writer considering ignoring the call), it also commonly means a simple "also." In that sense, it would mean that the writer thought of several things, of which "Maybe I'll ignore it" was one.
Here (again, with zero context), I get the sense of a fleeting thought that crossed the mind before the speaker decided to pick up the phone after all.
