I have an old (1939) BW glossy photo that's only 1" x 1.25". The photo quality is good, very sharp, just too tiny. What would be the best way to scan it to enlarge to a 5x7?
Normally, the idea is that to print double size, you scan at 600 dpi and print at 300 dpi (for 2x).
Or to print triple size, you scan at 900 dpi, and print at 300 dpi (for 3x). Anyway, that is the idea for printing enlarged copies.
However, prints really don't enlarge very well. One, prints are already enlarged, and two, print paper is simply only designed to be viewed by the eye, and they are NOT designed to be enlarged (film is designed to be enlarged). So reprinting original size is best, and good. 2x copies can be fair, and 3x is becoming objectionable, because prints really don't have the resolution to give. You can scan at 900 dpi, but the results just wont be that good (from prints). Scanning film enlarges pretty well larger, maybe 10x or 15x, but prints, not so much.
Old B&W contact prints (like yours, not enlarged) are an exception in some degree. These are sharper, and contain more fine detail. Triple size may be fair results after all.
Your 1x1.25 to 5x7 is 5.6x enlargement. That is 300x5.6 = 1680 dpi scan resolution to print 300 dpi. That is stretching it however. I wouldn't expect much result from more than 1200 dpi, if that. 1200 dpi would print 6x at 200 dpi. But it is still 6x.
I would scan it twice, at 1200 or 1600 dpi that you want, and also at 900 dpi for printing triple size that you may have to settle for, just in case.
There are photo service bureaus that will scan your film for you, and there are many scanners available that can support scanning transparencies or film. For example the Epson Perfection V600 http://amzn.to/2avk6wY . You'd need to make sure it can mount your film size.
It is best to scan at the native resolution of the scanner. Many times scanners have asymmetric horizontal and vertical resolutions. In that case pick the lowers of the two. In the case of the Epson listed above the resolution is 6400 x 9600 dpi, so scan at 6400 dpi. You may need other image editing software like PhotoShop or The Gimp to invert the scan image and adjust the levels for best exposure.
Beyond scanning at the scanner's highest native resolution, you can combine multiple scans made at 90º and 180º orientations to the original.
Although the problems are slightly different, the same solution will work here as well:
How to restore as much detail as possible to a scanned image of an inkjet-printed page?
What is the best way to remove texture from a scanned textured photo paper?