If I'm reading your post correctly, that $2000 or so is your emergency fund. That's insurance against an emergency. Insurance is expensive. You say you want it working for you? It is working for you: it's protecting you in case of an emergency.
Until you have reliable, regular passive income outside of a tax-advantaged account, keep your emergency fund in a fully liquid place - savings account of some sort. MMM doesn't maintain a large emergency fund - he also has a ton of money in taxable, and the distributions are regular and cover his expenses; in his case, he doesn't need that additional insurance. From your description, you do need it.
I have a bit of profit I was able to pull from my business recently and so have another $2000 or so that I could direct to some kind of investment vehicle.
To be honest, I'd add that to the savings account. Get yourself a few months' expenses in savings before you look into additional investing (beyond a 401(k) match).
So I’m considering keeping a little money in the savings account for maximum liquidity, but investing additional into an index fund or something to get it working for me while still being accessible in an emergency.
I'm a big fan of index funds, but
they do not fit the criteria you listed. If you take, for example, a stock index fund, and there's an emergency, it could very well be down 50% - right when you need the money. Stocks are not an appropriate investment for your emergency fund. Once you have a basic tier of your emergency fund, you can consider moving additional tiers of it into something like a short-term bond index fund and still be part of that.
I’d really like to accelerate the HELOC pay down and get it out of my life, so I could also make a big lump principal payment to the HELOC which could save me between a few hundred and a thousand bucks over the life of the loan, but then I don’t have the cash on hand any more, of course. And it only saves me a year or two of payments.
I'd calculate the benefit in terms of how much money it savings you in interest, not in how many checks you write.
Since the rate on the HELOC is close to what I might make on investments, maybe it’s a wash and I should just get the index fund started. And if the market does well, maybe the earnings on the investments will exceed the interest.
And maybe it loses money? What then?
So that’s where I’m leaning now—put it all in an investment vehicle. Any thoughts or advice? Also, does anyone have a good resource for long-term silver price forecasts? I could sell and invest it but I’d kick myself if it rises $20 by New Year’s!
First, no one has accurate long-term silver price forecasts. It's a largely speculative instrument. Folks trying to sell it will tell you it's poised to go up; folks looking to buy tell you it's on the way down. I avoid precious metals "investing" for many reasons - including the lack of real value to it and the ideological baggage that is usually attached to individuals thinking of it as an investment.
You could sell now and invest it - or shore up your emergency fund - and you say you'd kick yourself if it goes up $20/unit in a few months. What if you don't sell and it drops $20/unit?
Long story short: I'd sell the silver and keep the cash. Shore up the emergency fund first, then let's talk investing. Put simply, it doesn't sound to me like your financial situation will allow you to invest without taking unnecessary risk.