I have an early 2000s Litespeed frame that I'd like to build up but it has a 1" threaded headset. Would it be possible/cost effective to have the frame modded to modern head tube standards? At least 1 1/8 threadless.
Where would I find someone to do this?

While this is possible the cost and effort is not generally considered worth it. You would want to work with a frame builder who is fluent in TI (assuming that's the frame material as most Lightspeeds are that). In steel I would only consider this with also replacing the top and down tubes. But it's your bike and all you need is to find someone who says yes and can actually do the work well. Andy

A company like Ti Cycles in Portland could do the work, but it wouldn't be cheap.

I wouldn't encourage doing it since Titanium welds can be a bit finicky, and additional heat cycles at the place of the previous weld is probably a bad idea, and could lead to frame failure.

The easiest solution would be to install a 1" threadless fork plus shims to use standard 1 1/8" stems.

About 6 years ago I inquired with Dave Levy at Ti Cycles Fabrication about modding my Ti Cycles Softride frame to a 1-1/8" head tube, as well as tweaking the head tube angle slightly. While I won't quote you the price, and it may have been affected by the fact that he built the original frame, I thought it was quite reasonable. I didn't follow through with doing it though (the frame is still hanging in my shop).

As a point of reference, "head tube replacement" on a titanium frame is listed at $600 on Bilenky's website: Bilenky Cycle Works | Repairs. This seems more or less in line with what I'd expect from other builders willing to take on the challenge. If you really like everything about the frame except you want to use a modern fork with it, I'd say it's worth the investment, but if it's just any old titanium frame for you, you can find titanium frames on ebay that already take a 1 1/8 or tapered fork for less than that.

I found that in a search last night. Found another guy in Portland who quoted $400 on his website.
I'm probably not serious about this. I have a road and gravel bike already so I don't need anything else but was trying to brainstorm something to do with this frame.

You can get a threadless Columbus Minimal carbon for for $165. Add a threadless headless and stem (with shim as needed) and you should be good to go. It doesn't make sense to mod the frame in my opinion.

I agree that the frame mod idea doesn't make sense, I was just blue skying.
That fork would be great except it's for road tires. 25mm max. As it sits this frame can take up to 50mm tires which I think adds to its value.