I was confused for a moment, since the opening sentence and thread title talks about multimeter probes, then the third sentence talks about "standard probes" but it then becomes apparent you're talking about oscilloscope probes...
Anyhow:
Does anyone know what I'm talking about?
I assume you mean precision multimeter probes like the Fluke TL910 (aka Pomona 6341/6342/6275) or Probe Master 8050? Both of those have a pogo pin receptacle into which pogo pins are inserted to function as probes. The TL910 comes with both gold-plated pogo pins and rigid unplated stainless steel needles. The Probe Master used to come with both gold-plated pogo pins and rigid gold-plated stainless steel needles, but they no longer include the rigid needles (nor appear to sell them as spares anymore).
I frequently use standard probes with the spring clip removed and the ground lead replaced with a ground spring, so the pointed tip of the probe rests in a large via and the ground spring rests in any nearby ground via. This gets much better signal quality than using the ground lead, especially in power electronics projects. The probe will even balance this way under its own gravity, allowing me to set up many measurements hands free with very short ground connections.
I want to make a DIY probe that can rest on vias in a similar way. I'm envisioning a narrow PCB with an amplifier under a metal EMI shield and two contact points for contacting vias to measure them. To account for different via sizes and spacings, I'd like to have replaceable tips that I can bend into different shapes for different measurements: straight, with a small kink, or with a large kink (and short to minimize loop area or long for harder to reach parts of the board or to get more probes to a small area).
Sounds to me like you're basically trying make something mechanically similar to the Positioner Tip of the LeCroy
DX20-PT-KIT. (And knowing LeCroy's probe pricing, that probably cost thousands and thousands...)
My understanding is that multimeter probe tips are often made of tin coated steel but are occasionally gold coated
No. They're usually either brass plated in nickel or gold, unplated stainless steel, or gold-plated stainless steel. Tin tarnishes readily and would make a terrible plating.
is the same true of oscilloscope probes?
I believe they're the same as multimeter probes. The LeCroy Adjustable Tip tips are nitinol, according to the datasheet above.
Or is there a material that could be appropriate for use as a probe tip without any coating, so I could just buy a rod, cut it into lengths, then add points with a bench grinder?
Stainless steel, as seen above.
All the googling I've done has just gotten me places to buy complete probes, but I'm just looking for simple replaceable tips.
Pomona sells just the replacement tips for the TL910/6341/6342/6275 as model number 6354 for the complete set. (They used to sell the stainless needles alone as 6213 for short ones and 6214 for long ones, but that's seemingly discontinued.) In either case they're 0.040" diameter.
Also, take a look at LeCroy PK1-5MM-104. I don't know the shaft diameter, however.