I suggest that she "try" calling after lunch but she ignores me??
tried makes more sense then "try" tried is in the past tense but even so its hard to say
"Tried" is the past tense right enough, but the past tense is already there in "suggested". Try this sentence: "I suggested that she tried..." It doesn't quite work does it?
What's required here is actually the present tense, but in the subjunctive mood.
The subjunctive mood is dying out in English, but this is one example where it is still often used: following verbs of command or request. "Suggest" here is a verb of command (although it's a very mild command).
The subjunctive looks like the basic form of the verb. Here are some examples:
I inisist that he leave now.
They recommended that Jane appeal the sentence.
We suggest you be here tomorrow.
In British English, this rule is not quite so well-known, and many native speakers get it wrong. Instead, we sometimes use "should", like this: