Lanark Museum is a small local-history stop rather than a destination museum, but it gives useful context for the old county town. Nearby Castlegate is worth a look for St Nicholas Church and the Girnin Dug, a 19th-century rooftop dog statue with a suitably petty neighbourly backstory.
Treat this as a short town-centre add-on before or after New Lanark, not the main event.