Is a day trip to Paris from London worth it, and how do I go about it on a budget?

You could spend most of the day in one museum alone so a day trip sounds expensive. Touring Paris in a day is like trying to see NYC in a day. Try to allow more time. Better yet, just see London for that day (another world class city) and visit Paris some other time. The biggest expense would be the air fare. Cheap food? You can get ready cooked foods at deli sections in supermarkets. This is the best kept secret which is missed by many tourists. Quick is a chain of Belgian owned burger joints which you might like. The cheapest way to eat is to buy groceries from supermarkets and there is a lot of good tasting food (including canned goods) to be found. French law requires that all restaurants post the menu out front where you can stand on the sidewalk and read the selections (and the prices). I have heard that some Vietnamese restaurants have low prices, but go only if you really want to try Vietnamese food (an acquired taste).

No, a day trip isn't worth the expense. Atlantic sea raiders rowed to Paris taking time for plunder and rapine; all of which you never thought about. Are you on a designated victim list?