"To generate a lot of suggestions" saying
Solution 1:
"Brainstorm" seems appropriate. It's generally used in the context of a group, but there's nothing wrong with using it to describe one person's action.
In case you need a noun, ideation is the process of generating ideas.
"Throw out" can work, but it could also be interpreted as "discard" or "reject" if the context isn't clear.
You could also say, "Tim offered/proposed/generated a(n) array/stream of ideas" if "brainstorm" isn't casual enough.
As a side note, "random" implies that the ideas may be irrelevant, poor, out of context.