You might want to do some basic reading about beta decays. Wikipedia at the very least. Beta decay comes in two flavors, + and -. In both cases a neutron decays into a proton and either an electron and an antineutrino or a positron and a neutrino. The emitted electrons/positrons are sometimes called Beta radiation, and I assume this is what you're talking about.
Electrons and positrons are not remotely useful in inducing fission, which would be required to convert lead (atomic number 82) to gold (atomic number 79). Inducing beta decay in lead would also lead in the wrong direction, since converting an neutron to a proton increases the atomic number of a nucleus, rather than decreasing it.
ETA: I misspoke - that is, I was wrong about the details of $\beta$ + decay. Such a process will decrease atomic number by one, so inducing 3 successive instances of $\beta$ + decay would (in theory) produce gold. For about a microsecond.
Lead occurs in 3 isotopes, 206Pb, 207Pb, and 208Pb. A $\beta$ + decay will convert these to thallium, specifically, 206Tl, 207Tl and 208Tl. Each of these isotopes will decay via $\beta$ - decay back to lead with half-lives of 3-4 minutes. But let's say we can induce further $\beta$ + decay and convert the thallium to mercury before the atoms have a chance to decay.
This will produce 206Hg, 207Hg, and 208 Hg. Each of these isotopes will decay via $\beta$ - decay back to thallium, with a half-life of 3 to 42 minutes. But let's say we again induce $\beta$ + decay, and produce gold, specifically 206Au, 207Au and 208Au.
These gold isotopes will also decay via $\beta$ - decay back to mercury, but the half-life for these isotopes is small but greater than 300 nsec.
And, of course, neither electrons nor positrons will trigger beta decay of either variety, so beta radiation cannot possibly trigger transmutation.
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So the simple answer is no.
Equally important is the fact that the Philosopher's Stone had other uses beyond transmutation. The Stone's transformative powers arose from its power to perfect, thereby changing lower "base" (inferior) metals to the more "noble" metals such as silver and gold. As a result, the stone also served as an elixir of life, since it could be used to eliminate imperfections in a person, restoring health and extending lifetime. A body which does not die, of course, is clearly a more perfect body than one which does.
A source of beta radiation is not what you'd call healthy, and $\beta +$ decay produces positrons which annihilate with any nearby electrons to produce high-energy gamma radiation, which is really not healthy.
So on this count also, a source of beta radiation is not remotely a Philosopher's Stone.