I have just been given, by a neighbor, an old, rusting mountain bike. Well, its at least 5 years old and been sitting out in all weathers.

The chain is rusty, the crank rusty, cassette rusty.

I want to do it up for my granddaughter to use, also be a wee project for me. I would like to keep it as original as possible.

Any advice on how to deal with rust on a bike? I am not willing to spend too much money on it, but would love to spend time with my grandkid fixing it up....get her to take an active part.

Any and all advice would be greatly appreciated.

Peace

Toss the chain and get a new one.
Take the bike apart and soak the crank and cassette in mineral spirits. Or you can try scrubbing with a piece of foil dipped in baking soda and vinegar.

Is the frame steel and is there a lot of internal rust? That would be the decider as to whether the bike is worth cleaning up and using. If the frame is solid, then jimmie65's advice is a good starting point. Also disassemble, clean and relube the hubs, headset and bottom bracket. Replace the brake shoes, install all new cables and housing and clean and lube the derailleurs.

A lot of how much you will replace or just clean / refinish will depend on what grade of bike it is, and how much use you think it will get.

I have two 'free' bikes in my fleet, one was a vintage Bridgestone that had spent 18 years under a house and the other was my wife's department-store MTB that I found behind a dumpster.

The Bridgestone was a nice, if not top-line bike when it was new, and I was looking for a road bike for cross-training and recovery (injured middle-distance runner) That bike gets used ~ 3 times a week, 20-30 miles at a time. I even did my first two centuries on it. That bike got a full rebuild, down to the frame, all the bearings were disassembled, cleaned and repacked. New cables, tape, saddle, tires and some upgrades, about $200, and several weeks worth of evenings. I probably could have bought a newer, lighter, better performing bike for the $200, but it was my dad's bike, so it has some personal value to me.

My wife's bike is ~5 years old or so, and was probably $75-$100 when it was new. It maybe gets used a dozen times a year, less then 5 miles each time. There was no rust on the frame, so it just got some polish. The chain was rusted solid, so it got swapped with a used one from my spares; the torn seat replaced with a $5 take-off from the bike shop. All the mechs were inspected and given a coat of either black or silver paint, lubed and adjusted.
A rebuild to the extent I did the Bridgestone would have exceeded the value of bike and would have been overkill for the amount of use it sees.

Rust on the gears is pretty easy to take care of, especially the rear cogs. I just spray the cassette with a lube like WD40, and give it a good scrub with a wire-bristle brush to get the gunk off, after that, riding the bike will get the rust off the teeth in quick order.
A lot of the black finished parts like headsets, and the like, especially on lower end bikes can be visually cleaned up by applying some penetrating oil and rubbing vigorously with a shop rag. The rust in most of these cases is a very thin layer, an comes off easily.
Chrome parts can be cleaned up using the aluminum foil and vinegar / water method. This works well, however, anywhere there's a rust 'pit' in chrome, it will polish out to the bare metal, but it takes a lot of elbow grease.

Post a picture of the bike, if you can, and we can give you a better idea of what it will take, and what issues you might run in to.

So much depends on exactly what bike you have and the quality of the components. If it is a cheap discount store bike, that was never very good, and the rust is severe enough, you do have to consider how much time or effort you want to expend just to end up with an even worse bike than it was when new.

John