I’d be grateful if you could take a look when you’ve got time sometime.
Can you explain the にでも part? Is it a way to make 時間のある時 more roundabout? like saying, “if you might even have the slightest bit of time” rather than “if you have time” ?
The first sentence sounds natural but also a bit unfriendly. The nuance is “you need to take a look at it when you have spare time” or “once you find a free time, you should check it out”. The speaker exactly points at the time when the listener should take a look at it.
Inserting でも in the second sentence functions as “for example, your free time, maybe?”, so the tone of the whole sentence is softened. If the listener has a spare time, “checking it up” is just one of the many things to do. The listener isn’t expected to give the highest priority to the speaker’s request. And the timing is totally up to the listener. The speaker merely makes a suggestion from other many possible time slots. Therefore, the second sentence is less demanding and more polite.
Here are other example sentences with でも in the same grammatical structure.
駅前に新しいレストランができた。今度、仕事帰りにでも寄ってみようかな。(えきまえにあたらしい… こんど、しごとがえりにでもよってみようかな) = I found a new restaurant in front of the restaurant. I’m thinking of visiting there, say… after work in the near future, maybe?”
去年の夏休みには家族で沖縄に旅行した。今年は北海道にでも行こう。(きょねんのなつやすみはかぞくでおきなわにりょこうした。ことしはほっかいどうにでもいこう) = I visited Okinawa with my family on vacation last summer. Maybe Hokkaido this year.
“It’s just one of the many opinions” is the meaning of でも. The original lesson sentence is a dialogue between the speaker and the listener, but you can also use でも when you express your own undecided plans/options.