JavaScript, 217 215 bytes
a="a=q;a=a.replace('q',uneval(a));for(b=c='a';d=a.split(c),c=d[Math.random()*~-d.length+1|0][0];b+=c);alert(b)";a=a.replace('q',uneval(a));for(b=c='a';d=a.split(c),c=d[Math.random()*~-d.length+1|0][0];b+=c);alert(b) Note that this uses uneval, which is only supported by Firefox. Sample runs:
a=ale(a.lend[Ma=d[Macepla.ler(b+=c)b=q;fom(a=q;a=dort(b+1|0],c);a.lit(a)
at(c=c;d[0],c=q;ath+1|0][0];dorerac=ac=d[Ma),c;)*~-d[Ma=alenepl(b+=ac=c;a=c;d[2];d.re(c;fom()
a="a[0],und=d=a)
angt(b),und=d.l(b=a)
a)
ale(a.rth.revanepleplit(b)
ac);fore(b)*~-d.r(b+1|0];fora';a)*~-d.splalith+=dorth+=c=";ath+=a.length+=';ale(b)
a.r(b=c=a)b+1|0],und[0][0];d.splerath.spleneva)";ath.r(ceneplith+=d=aceple(c;)*~-d=';ala';)b='ac;fom(b=c;a.ler(b=d=d[Ma.rt(c=cendor()*~-d='a=";ac;a.spla)b=ceva=';a=d.rt(angt(alength+1|0],c;angt()
al(ac=dorth+1|0][0][0][0][Ma.split()
As you can see, it's mostly gibberish, but that is to be expected ;) The OP has created a JSFiddleJSFiddle which demonstrates that the chance of an output being syntactically valid JS is about 6.3%.
If self-reading functions were allowed, this could be 78 bytes of ES6:
f=(c="f",p=("f="+f).split(c),q=p[Math.random()*~-p.length+1|0][0])=>q?c+f(q):c
Very, very rarely, this outputs syntactically valid JS:
f=>e?c+f():c
f=>e?c=>engt():c
f=>e?c=(e):c
f=>e?c=>e=>ength.split():c
f=p=>q?c+f():c
f(q).sp=",p[Mat(q?c=(),plith.lith.sp.sp[0]).lendom().lith+f=>q=p.lendom(",p=p=>q?c+f():c
f(q),q?c=(c=(q)*~-p[0]):c
f().random(),q?c=(c=p[0]):c
f=>q?c=(q="+f"+f).rath.split(c):c
f="+1|0])=().lith.rat()*~-p=>q?c=p[Mat(c=",q?c=p.rath.splendom()*~-plength.splith.lendom(c):c
My favorite of the function names it's created is .splendom() (split + length + random)
